**2.1 Charge**

When certain objects are rubbed together, they undergo a dramatic change. Whereas before these objects exerted no noticeable forces on their environment, they now do. For example, if you hold one of the objects near a small piece of paper, the piece of paper may jump up towards and attach itself to the object. Put this in perspective: *the entire Earth* is exerting a gravitational pull on the piece of paper, but a comparatively small object is able to exert a force big enough to overcome this pull (Arons, 1996).

If we take the standard example of rubber rods rubbed with cat fur, and glass rods rubbed with silk, we observe that all rubber rods repel each other as do all glass rods, while all rubber rods attract all glass rods. It turns out that all charged objects ever experimented on either

and for the field due to a point charge *Q*,

*r*

φ d*Er* d*E*

*P*

**2.3 An infinite line charge**

*z*

*z*

symmetrically placed segments.

cancel, leaving only the *r* component:

thus an infinite sum, given by the integral

φ

d*z*

*r*

 *<sup>E</sup>* <sup>=</sup> <sup>1</sup> 4*π�*<sup>0</sup>

objects generally are affected by other charges, for example through polarization.

*Q*

*z*

*P*

(a) (b)

*z*

. (4)

. (5)

*<sup>r</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> . (6)

d*z*

d*z*

*z*

Finally, experiments show that Coulomb's Law obeys the superposition principle; that is to say, the force exerted between two point-like charged objects is unaffected by the presence or absence of other point-like charged objects, and the net electrostatic force on a point-like object is found by adding all individual electrostatic forces acting on it. Of course, macroscopic

Current-Carrying Wires and Special Relativity 5

*r*

Fig. 1. Linear charges: (a) field due to a small segment of length d*l*, (b) net field due to two

Imagine an infinitely long line of uniform linear charge density *λ*. Take a segment of length d*z*, a horizontal distance *z* from point *P* which has a perpendicular distance *r* to the line charge.

4*π�*0(*r*<sup>2</sup> + *z*2)

4*π�*0(*r*<sup>2</sup> + *z*2)

d*z* sin *φ*

A second segment of the same length d*z* a distance *z* from *P* (see Fig. 1b) gives rise to an electric field of the same magnitude, but pointing in a different direction. The *z* components

<sup>d</sup>*Er* <sup>=</sup> *<sup>λ</sup>*d*<sup>z</sup>* sin *<sup>φ</sup>*

To find the net field at *P*, we add the contributions due to all line segments. This net field is

 ∞ −∞

*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>* 4*π�*<sup>0</sup>

By Coulomb's Law, the magnitude of the electric field at *P* due this line segment is

<sup>d</sup>*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>*d*<sup>z</sup>*

*<sup>r</sup>*<sup>2</sup> *<sup>r</sup>*ˆ. (3)

behave like a rubber rod, or like a glass rod. This leads us to postulate that there only two types of charge state, which we call positive and negative charge for short.

As it turns out, there are also two types of charge: a positive charge as found on protons, and a negative charge as found on electrons. In this chapter, a wire will be modeled as a line of positively charged ions and negatively charged electrons; these two charge states come about through separation of one type of charge (due to electrons) from previously neutral atoms. However, the atoms themselves were electrically neutral due to equal amounts of the type of charge due to the protons in the nucleus, and the type of charge due to electrons.

Charged objects noticeably exert forces on each other when there is some distance between them. Since the 19th century, we have come to describe this behaviour in terms of electric fields. The idea is that one charged object generates a field that pervades the space around it; this field, in turn, acts on the second object.

### **2.2 Coulomb's Law**

Late in the 18th century, Coulomb used a torsion balance to show that two small charged spheres exert a force on each other that is proportional to the inverse square of the distance between the centres of the spheres, and acts along the line joining the centres (Shamos, 1987a). He also showed that, as a consequence of this inverse square law, all charge on a conductor must reside on the surface. Moreover, by the shell theorem (Wikipedia, 2011) the forces between two perfectly spherical hollow shells are exactly as if all the charge were concentrated at the centre of each sphere. This situation is very closely approximated by two spherical insulators charged by friction, the deviation arising from a very small polarisation effect.

Coulomb also was the first person to quantify charge. For example, having completed one measurement, he halved the charge on a sphere by bringing it in contact with an identical sphere. When returning the sphere to the torsion balance, he measured that the force between the spheres had halved (Arons, 1996). When he repeated this procedure with the other sphere in the balance, the force between the spheres became one-quarter of its original value.

In modern notation, Coulomb thus found the law that bears his name: the electrostatic force *FE* between two point-like objects a distance *r* apart, with charge *Q* and *q* respectively, is given by

$$
\vec{F}\_E = \frac{1}{4\pi\epsilon\_0} \frac{Qq}{r^2} \hat{r}.\tag{1}
$$

In SI units, the constant of proportionality is given as 1/4*π�*<sup>0</sup> for convenience in calculations. The constant *�*<sup>0</sup> is called the permittivity of vacuum.

It is often useful to define the charge per unit length, called the linear charge density (symbol: *λ*); the charge per unit (surface) area, symbol: *σ*; and the charge per unit volume, symbol *ρ*.

We are now in a position to define the electric field *E* mathematically. The electric field is defined as the ratio of the force on an object and its charge. Hence, generally,

$$
\vec{E} \equiv \frac{\vec{F}\_E}{q},
\tag{2}
$$

and for the field due to a point charge *Q*,

$$
\vec{E} = \frac{1}{4\pi\epsilon\_0} \frac{Q}{r^2} \hat{r}.\tag{3}
$$

Finally, experiments show that Coulomb's Law obeys the superposition principle; that is to say, the force exerted between two point-like charged objects is unaffected by the presence or absence of other point-like charged objects, and the net electrostatic force on a point-like object is found by adding all individual electrostatic forces acting on it. Of course, macroscopic objects generally are affected by other charges, for example through polarization.

#### **2.3 An infinite line charge**

2 Will-be-set-by-IN-TECH

behave like a rubber rod, or like a glass rod. This leads us to postulate that there only two

As it turns out, there are also two types of charge: a positive charge as found on protons, and a negative charge as found on electrons. In this chapter, a wire will be modeled as a line of positively charged ions and negatively charged electrons; these two charge states come about through separation of one type of charge (due to electrons) from previously neutral atoms. However, the atoms themselves were electrically neutral due to equal amounts of the type of

Charged objects noticeably exert forces on each other when there is some distance between them. Since the 19th century, we have come to describe this behaviour in terms of electric fields. The idea is that one charged object generates a field that pervades the space around it;

Late in the 18th century, Coulomb used a torsion balance to show that two small charged spheres exert a force on each other that is proportional to the inverse square of the distance between the centres of the spheres, and acts along the line joining the centres (Shamos, 1987a). He also showed that, as a consequence of this inverse square law, all charge on a conductor must reside on the surface. Moreover, by the shell theorem (Wikipedia, 2011) the forces between two perfectly spherical hollow shells are exactly as if all the charge were concentrated at the centre of each sphere. This situation is very closely approximated by two spherical insulators charged by friction, the deviation arising from a very small polarisation effect.

Coulomb also was the first person to quantify charge. For example, having completed one measurement, he halved the charge on a sphere by bringing it in contact with an identical sphere. When returning the sphere to the torsion balance, he measured that the force between the spheres had halved (Arons, 1996). When he repeated this procedure with the other sphere

In modern notation, Coulomb thus found the law that bears his name: the electrostatic force

*FE* between two point-like objects a distance *r* apart, with charge *Q* and *q* respectively, is given

In SI units, the constant of proportionality is given as 1/4*π�*<sup>0</sup> for convenience in calculations.

It is often useful to define the charge per unit length, called the linear charge density (symbol: *λ*); the charge per unit (surface) area, symbol: *σ*; and the charge per unit volume, symbol *ρ*.

*Qq*

*<sup>r</sup>*<sup>2</sup> *<sup>r</sup>*ˆ. (1)

*E* mathematically. The electric field is

*<sup>q</sup>* , (2)

in the balance, the force between the spheres became one-quarter of its original value.

 *FE* <sup>=</sup> <sup>1</sup> 4*π�*<sup>0</sup>

defined as the ratio of the force on an object and its charge. Hence, generally,

 *<sup>E</sup>* <sup>≡</sup> *FE*

The constant *�*<sup>0</sup> is called the permittivity of vacuum.

We are now in a position to define the electric field

types of charge state, which we call positive and negative charge for short.

charge due to the protons in the nucleus, and the type of charge due to electrons.

this field, in turn, acts on the second object.

**2.2 Coulomb's Law**

by

Imagine an infinitely long line of uniform linear charge density *λ*. Take a segment of length d*z*, a horizontal distance *z* from point *P* which has a perpendicular distance *r* to the line charge. By Coulomb's Law, the magnitude of the electric field at *P* due this line segment is

$$\mathrm{d}E = \frac{\lambda \mathrm{d}z}{4\pi\epsilon\_0(r^2 + z^2)}.\tag{4}$$

A second segment of the same length d*z* a distance *z* from *P* (see Fig. 1b) gives rise to an electric field of the same magnitude, but pointing in a different direction. The *z* components cancel, leaving only the *r* component:

$$\mathrm{d}E\_{\mathrm{F}} = \frac{\lambda \mathrm{d}z \sin \phi}{4\pi \epsilon\_0 (r^2 + z^2)}. \tag{5}$$

To find the net field at *P*, we add the contributions due to all line segments. This net field is thus an infinite sum, given by the integral

$$E = \frac{\lambda}{4\pi\epsilon\_0} \int\_{-\infty}^{\infty} \frac{\mathbf{d}z \sin\phi}{r^2 + z^2}. \tag{6}$$

each gives rise to an electric field of magnitude

and constants:

hence

is given as

as expected.

<sup>d</sup>*<sup>E</sup>* <sup>=</sup> *<sup>σ</sup>R*d*<sup>φ</sup>*

Current-Carrying Wires and Special Relativity 7

The net field at any point *P* follows from superposition. We use a righthanded Cartesian coordinate system where the positive *y*-axis points up and the positive *z*-axis points out of the page. When comparing the contributions from the right half of the cylinder to the electric field with those from the left half, it is clear by symmetry that the *y*-components are equal and add,

> <sup>d</sup>*Ey* <sup>=</sup> *<sup>σ</sup><sup>R</sup> π�*0

The integrand in (12) contains 3 variables, *r*, *φ*, and *θ*. We may write *r* and cos *θ* in terms of *φ*

*R*<sup>2</sup> + *y*<sup>2</sup>

When entering the integral into the Mathematica online integrator (2011), the antiderivative

which is admittedly ugly, but not difficult to use. Since arctan is an odd function, the first two

*<sup>y</sup>*<sup>0</sup> cos *<sup>x</sup>*/2 <sup>−</sup> *<sup>R</sup>* sin *<sup>x</sup>*/2

Substitution eventually yields that the value of the integral is *π*/*y*0. Hence Equation (14) gives

*π y*0

*<sup>E</sup>* <sup>=</sup> *<sup>σ</sup><sup>R</sup> π�*0

*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>* 2*π�*0*y*<sup>0</sup>

 *π*/2 −*π*/2

<sup>0</sup> − 2*y*0*R* sin *φ*

arctan( *<sup>y</sup>*<sup>0</sup> sin *<sup>x</sup>*/2−*<sup>R</sup>* cos *<sup>x</sup>*/2 *<sup>y</sup>*<sup>0</sup> cos *<sup>x</sup>*/2−*<sup>R</sup>* sin *<sup>x</sup>*/2 )

2*y*<sup>0</sup>

+ *x* 2 

*y*<sup>0</sup> − *R* sin *φ*

cos *θ r*

*R*<sup>2</sup> + *y*<sup>2</sup>

along the direction *AP* pointing away from the line charge, as shown in Figure 2.

while the *x*-components are equal and subtract to yield zero. Hence

 *π*/2 −*π*/2

*<sup>r</sup>* <sup>=</sup> (*<sup>R</sup>* cos *<sup>φ</sup>*)<sup>2</sup> + (*<sup>R</sup>* sin *<sup>φ</sup>* <sup>−</sup> *<sup>y</sup>*0)<sup>2</sup> <sup>=</sup>

 *π*/2 −*π*/2

+

arctan *<sup>y</sup>*<sup>0</sup> sin *<sup>x</sup>*/2 <sup>−</sup> *<sup>R</sup>* cos *<sup>x</sup>*/2

*E* = 2

*<sup>E</sup>* <sup>=</sup> *<sup>σ</sup><sup>R</sup> π�*0

*<sup>y</sup>*<sup>0</sup> cos *<sup>x</sup>*/2−*<sup>R</sup>* sin *<sup>x</sup>*/2 )

cos *θ* = *<sup>y</sup>*0−*<sup>R</sup>* sin *<sup>φ</sup> r*

<sup>−</sup> arctan( *<sup>R</sup>* cos *<sup>x</sup>*/2−*y*<sup>0</sup> sin *<sup>x</sup>*/2

2*y*<sup>0</sup>

terms are identical, and the antiderivative simplifies to

1 *y*0 

for the electric field *E* outside the hollow cylinder:

which, defining *λ* = *σ* · 2*πR*, simplifies to

<sup>2</sup>*π�*0*<sup>r</sup>* (11)

<sup>0</sup> − 2*Ry*<sup>0</sup> sin *φ*

+ *x* 2*y*<sup>0</sup>

*π*/2

−*π*/2 .

, (15)

, (16)

d*φ* (12)

d*φ*. (14)

  *π*/2

−*π*/2

,

; (13)

The integral in (6) contains two variables, *z* and *φ*; we must eliminate either. It can be seen from Fig. 1a that

$$\sin \phi = \frac{\text{dE}\_r}{\text{dE}} = \frac{r}{(r^2 + z^2)^{1/2}},\tag{7}$$

which allows us to eliminate *φ*, yielding

$$E = \frac{\lambda r}{4\pi\varepsilon\_0} \int\_{-\infty}^{\infty} \frac{\mathrm{d}z}{(r^2 + z^2)^{3/2}}.\tag{8}$$

The antiderivative is readily found manually, by online integrator, or from tables; the integration yields

$$\int\_{-\infty}^{\infty} \frac{\mathrm{d}z}{(r^2 + z^2)^{3/2}} = \frac{z}{r^2(r^2 + z^2)^{1/2}}\Big|\_{-\infty}^{\infty} = \frac{2}{r^2}.\tag{9}$$

Hence, the electric field due to an infinity linear charge at a distance *r* from the line charge is given by

$$E = \frac{\lambda}{2\pi\epsilon\_0 r}.\tag{10}$$

#### **2.4 Electric field due to a uniformly charged hollow cylinder**

Consider an infinitely long, infinitely thin hollow cylinder of radius *R*, with uniform surface charge density *σ*. A cross sectional view is given in Figure 2. What is the electric field at a point *P*, a distance *y*<sup>0</sup> from the centre of the cylinder axis? By analogy with the shell theorem,

Fig. 2. Uniformly charged hollow cylinder of radius *R*, with auxiliary variables defined.

one might expect that the answer is the same as if all the charge were placed at the central axis. For an infinite cylinder, this turns out to be true. Think of the hollow cylinder as a collection of infinitely many parallel infinitely long line charges arranged in a circular pattern. If the angular width of each line charge is d*φ*, then each has linear charge density *σR*d*φ*; by (10), each gives rise to an electric field of magnitude

$$\mathrm{d}E = \frac{\sigma R \mathrm{d}\phi}{2\pi\epsilon\_0 r} \tag{11}$$

along the direction *AP* pointing away from the line charge, as shown in Figure 2.

The net field at any point *P* follows from superposition. We use a righthanded Cartesian coordinate system where the positive *y*-axis points up and the positive *z*-axis points out of the page. When comparing the contributions from the right half of the cylinder to the electric field with those from the left half, it is clear by symmetry that the *y*-components are equal and add, while the *x*-components are equal and subtract to yield zero. Hence

$$E = 2\int\_{-\pi/2}^{\pi/2} \mathbf{d}E\_y = \frac{\sigma \mathbf{R}}{\pi \epsilon\_0} \int\_{-\pi/2}^{\pi/2} \frac{\cos \theta}{r} \mathbf{d}\phi \tag{12}$$

The integrand in (12) contains 3 variables, *r*, *φ*, and *θ*. We may write *r* and cos *θ* in terms of *φ* and constants:

$$\begin{cases} r = \sqrt{(R\cos\phi)^2 + (R\sin\phi - y\_0)^2} = \sqrt{R^2 + y\_0^2 - 2Ry\_0\sin\phi} \\ \cos\theta = \frac{y\_0 - R\sin\phi}{r} \end{cases} \tag{13}$$

hence

4 Will-be-set-by-IN-TECH

The integral in (6) contains two variables, *z* and *φ*; we must eliminate either. It can be seen

<sup>d</sup>*<sup>E</sup>* <sup>=</sup> *<sup>r</sup>*

d*z*

*r*2(*r*<sup>2</sup> + *z*2)1/2

*r*

*R*

φ

*A*

 ∞

−∞

<sup>=</sup> <sup>2</sup>

 ∞ −∞

The antiderivative is readily found manually, by online integrator, or from tables; the

Hence, the electric field due to an infinity linear charge at a distance *r* from the line charge is

Consider an infinitely long, infinitely thin hollow cylinder of radius *R*, with uniform surface charge density *σ*. A cross sectional view is given in Figure 2. What is the electric field at a point *P*, a distance *y*<sup>0</sup> from the centre of the cylinder axis? By analogy with the shell theorem,

*P*

*x*

Fig. 2. Uniformly charged hollow cylinder of radius *R*, with auxiliary variables defined.

one might expect that the answer is the same as if all the charge were placed at the central axis. For an infinite cylinder, this turns out to be true. Think of the hollow cylinder as a collection of infinitely many parallel infinitely long line charges arranged in a circular pattern. If the angular width of each line charge is d*φ*, then each has linear charge density *σR*d*φ*; by (10),

*y*

*z*

θ

*y*0

*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>* 2*π�*0*r*

(*r*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*2)1/2 , (7)

(*r*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*2)3/2 . (8)

. (10)

*<sup>r</sup>*<sup>2</sup> . (9)

sin *<sup>φ</sup>* <sup>=</sup> <sup>d</sup>*Er*

*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup><sup>r</sup>* 4*π�*<sup>0</sup>

d*z*

(*r*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*2)3/2 <sup>=</sup> *<sup>z</sup>*

from Fig. 1a that

integration yields

given by

which allows us to eliminate *φ*, yielding

 ∞ −∞

**2.4 Electric field due to a uniformly charged hollow cylinder**

$$E = \frac{\sigma R}{\pi \epsilon\_0} \int\_{-\pi/2}^{\pi/2} \frac{y\_0 - R \sin \phi}{R^2 + y\_0^2 - 2y\_0 R \sin \phi} \mathbf{d}\phi. \tag{14}$$

When entering the integral into the Mathematica online integrator (2011), the antiderivative is given as

$$\left. \frac{-\arctan\left(\frac{R\cos x/2 - y\_0 \sin x/2}{y\_0 \cos x/2 - R\sin x/2}\right)}{2y\_0} + \frac{\arctan\left(\frac{y\_0 \sin x/2 - R\cos x/2}{y\_0 \cos x/2 - R\sin x/2}\right)}{2y\_0} + \frac{x}{2y\_0} \right|\_{-\pi/2}^{\pi/2}$$

which is admittedly ugly, but not difficult to use. Since arctan is an odd function, the first two terms are identical, and the antiderivative simplifies to

$$\frac{1}{y\_0} \left[ \arctan \left( \frac{y\_0 \sin x / 2 - R \cos x / 2}{y\_0 \cos x / 2 - R \sin x / 2} \right) + \frac{x}{2} \right] \Big|\_{-\pi/2}^{\pi/2}$$

Substitution eventually yields that the value of the integral is *π*/*y*0. Hence Equation (14) gives for the electric field *E* outside the hollow cylinder:

$$E = \frac{\sigma \mathcal{R}}{\pi \epsilon\_0} \frac{\pi}{y\_0} \,, \tag{15}$$

.

which, defining *λ* = *σ* · 2*πR*, simplifies to

$$E = \frac{\lambda}{2\pi\epsilon\_0 y\_0},\tag{16}$$

as expected.

#### **2.5 Electric field due to a uniformly charged cylinder**

It follows from (16) that for any cylindrical charge distribution of radius *R* that is a function of *r* only, i.e., *ρ* = *ρ*(*r*), the electric field for *r* > *R* is given by

$$E = \frac{\lambda}{2\pi\epsilon\_0 r},\tag{17}$$

Just as Coulomb was able to abstract from a charged sphere to a point charge, the effect of a current can be abstracted to a steady "point-current" of length d*l*. (Note that a single moving point charge does not constitute a *steady* point-current.) In fact, there is a close analogy between the electric field due to a line of static charges and the magnetic field due to a line segment of moving charges – i.e., a steady linear current. The Biot-Savart law states that the

Current-Carrying Wires and Special Relativity 9

where *μ*<sup>0</sup> is a constant of proportionality called the permeability of vacuum, *I* is the current, d*z* is the length of an infinitesimal line segment, *φ* is the angle between the wire and the line connecting the segment to point *P*, the length of which is *R*; see Figure 3. Maxwell (1865) showed that *μ*<sup>0</sup> and *�*<sup>0</sup> are related; their product is equal to 1/*c*2, where *c* is the speed of light

*R*

φ

d*z*

*<sup>B</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *<sup>I</sup>* 2*π*

distance from the centre of the wire, varies with the distance *r* as

direction of the magnetic field is out of the page.

which has the exact same form as (6).

**4.1 Relativity in Newtonian mechanics**

*z*

Fig. 3. The Biot-Savart law: magnetic field due to a small segment carrying a current *I*. The

The magnetostatic force at point *P* due to an infinitely long straight current-carrying wire is

 ∞ −∞

Because the current distribution must have radial symmetry, all conclusions reached from (6) can be applied here. Thus, the magnetic field due to a steady current *I* in an infinitely long wire, hollow cylinder, or solid cylinder where the current density only depends on the

*<sup>B</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *<sup>I</sup>*

Newton's laws of motion were long assumed to be valid for all inertial reference frames. In Newton's model, an observer in one reference frame measures the position *x* of an object at

d*z* sin *φ*

d*z* sin *φ*

*r*

*P*

*B*

*<sup>R</sup>*<sup>2</sup> , (23)

*<sup>r</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> , (24)

<sup>2</sup>*π<sup>r</sup>* (25)

<sup>d</sup>*<sup>B</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *<sup>I</sup>* 2*π*

magnetic field at a point *P* due to a steady point current is given by

in vacuum.

then

outside the wire.

**4. Special relativity**

where the linear charge density *λ* is equal to the volume charge density *ρ* integrated over the radial and polar coordinates.

### **3. Magnetic fields and current-carrying wires**

#### **3.1 Current**

The flow of charge is called current. To be more precise, define a cross sectional area *A* through which a charge d*Q* flows in a time interval d*t*. The current *I* through this area is defined as

$$I \equiv \frac{\mathbf{d}Q}{\mathbf{d}t}.\tag{18}$$

It is often convenient to define a current density *J*, which is the current per unit cross sectional area *A*:

$$\mathbf{J} \equiv \mathbf{I}/\mathbf{A}.\tag{19}$$

A steady current flowing through a homogeneous wire can be modeled as a linear charge density *λ* moving at constant drift speed *vd*. In that case, the total charge flowing through a cross sectional area in a time interval Δ*t* is given by *λvd*Δ*t*, and

$$I = \lambda v\_d.\tag{20}$$

#### **3.2 Magnetic field due to a linear current**

In this chapter, we will only concern ourselves with magnetic effects due to straight current-carrying wires. Oersted found experimentally that a magnet (compass needle) gets deflected when placed near a current-carrying wire (Shamos, 1987b). As in electrostatics, we model this behaviour by invoking a field: the current in the wire creates a magnetic field *B* that acts on the magnet.

In subsequent decades, experiments showed that moving charged objects are affected by magnetic fields. The magnetostatic force (so called because the source of the magnetic field is steady; it is also often called the Lorentz force) is proportional to the charge *q*, the speed *v*, the field *B*, and the sine of the angle *φ* between *v* and *B*; it is also perpendicular to *v* and *B*. In vector notation,

$$
\vec{F}\_m = q\vec{v} \times \vec{B};
\tag{21}
$$

in scalar notation,

$$F\_m = qvB\sin\phi.\tag{22}$$

As a corollary, two parallel currents exert a magnetostatic force on each other, as the charges in each wire move in the magnetic field of the other wire.

6 Will-be-set-by-IN-TECH

It follows from (16) that for any cylindrical charge distribution of radius *R* that is a function

*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>* 2*π�*0*r*

where the linear charge density *λ* is equal to the volume charge density *ρ* integrated over the

The flow of charge is called current. To be more precise, define a cross sectional area *A* through which a charge d*Q* flows in a time interval d*t*. The current *I* through this area is defined as

*<sup>I</sup>* <sup>≡</sup> <sup>d</sup>*<sup>Q</sup>*

It is often convenient to define a current density *J*, which is the current per unit cross sectional

A steady current flowing through a homogeneous wire can be modeled as a linear charge density *λ* moving at constant drift speed *vd*. In that case, the total charge flowing through a

In this chapter, we will only concern ourselves with magnetic effects due to straight current-carrying wires. Oersted found experimentally that a magnet (compass needle) gets deflected when placed near a current-carrying wire (Shamos, 1987b). As in electrostatics, we model this behaviour by invoking a field: the current in the wire creates a magnetic field *B*

In subsequent decades, experiments showed that moving charged objects are affected by magnetic fields. The magnetostatic force (so called because the source of the magnetic field is steady; it is also often called the Lorentz force) is proportional to the charge *q*, the speed *v*, the field *B*, and the sine of the angle *φ* between *v* and *B*; it is also perpendicular to *v* and *B*. In

*Fm* <sup>=</sup> *<sup>q</sup>*�*<sup>v</sup>* <sup>×</sup> �

As a corollary, two parallel currents exert a magnetostatic force on each other, as the charges

�

in each wire move in the magnetic field of the other wire.

, (17)

<sup>d</sup>*<sup>t</sup>* . (18)

*J* ≡ *I*/*A*. (19)

*I* = *λvd*. (20)

*B*; (21)

*Fm* = *qvB* sin *φ*. (22)

**2.5 Electric field due to a uniformly charged cylinder**

**3. Magnetic fields and current-carrying wires**

radial and polar coordinates.

**3.1 Current**

area *A*:

of *r* only, i.e., *ρ* = *ρ*(*r*), the electric field for *r* > *R* is given by

cross sectional area in a time interval Δ*t* is given by *λvd*Δ*t*, and

**3.2 Magnetic field due to a linear current**

that acts on the magnet.

vector notation,

in scalar notation,

Just as Coulomb was able to abstract from a charged sphere to a point charge, the effect of a current can be abstracted to a steady "point-current" of length d*l*. (Note that a single moving point charge does not constitute a *steady* point-current.) In fact, there is a close analogy between the electric field due to a line of static charges and the magnetic field due to a line segment of moving charges – i.e., a steady linear current. The Biot-Savart law states that the magnetic field at a point *P* due to a steady point current is given by

$$\mathbf{d}B = \frac{\mu\_0 I}{2\pi} \frac{\mathbf{d}z \sin\phi}{R^2},\tag{23}$$

where *μ*<sup>0</sup> is a constant of proportionality called the permeability of vacuum, *I* is the current, d*z* is the length of an infinitesimal line segment, *φ* is the angle between the wire and the line connecting the segment to point *P*, the length of which is *R*; see Figure 3. Maxwell (1865) showed that *μ*<sup>0</sup> and *�*<sup>0</sup> are related; their product is equal to 1/*c*2, where *c* is the speed of light in vacuum.

Fig. 3. The Biot-Savart law: magnetic field due to a small segment carrying a current *I*. The direction of the magnetic field is out of the page.

The magnetostatic force at point *P* due to an infinitely long straight current-carrying wire is then

$$B = \frac{\mu\_0 I}{2\pi} \int\_{-\infty}^{\infty} \frac{\mathrm{d}z \sin\phi}{r^2 + z^2} \,\mathrm{'}\tag{24}$$

which has the exact same form as (6).

Because the current distribution must have radial symmetry, all conclusions reached from (6) can be applied here. Thus, the magnetic field due to a steady current *I* in an infinitely long wire, hollow cylinder, or solid cylinder where the current density only depends on the distance from the centre of the wire, varies with the distance *r* as

$$B = \frac{\mu\_0 I}{2\pi r} \tag{25}$$

outside the wire.

#### **4. Special relativity**

#### **4.1 Relativity in Newtonian mechanics**

Newton's laws of motion were long assumed to be valid for all inertial reference frames. In Newton's model, an observer in one reference frame measures the position *x* of an object at various times *t*. An observer in a second reference frame moves with speed *v* relative to the first frame, with identical, synchronized clocks and metre sticks. Time intervals and lengths are assumed to be same for both observers.

The second observer sees the first observer move away at speed *v*. The distance between the two observers at a time *t* � is given by *vt*� . Hence, the second observer can use the measurements of the first observer, provided the following changes are made:

$$\mathbf{x}' = \mathbf{x} - vt\tag{26}$$

$$t'=t\tag{27}$$

For linear transformations, the third and fifth terms are zero. Hence we obtain:

2

2

<sup>+</sup> <sup>2</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�*∂t*�

Current-Carrying Wires and Special Relativity 11

<sup>+</sup> <sup>2</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�*∂t*�

> 2 <sup>−</sup> *<sup>c</sup>*<sup>2</sup>

To retain the wave equation (30), it is clear that the right-hand side of this equation must be zero while the terms in square brackets on the left-hand side must be equal. This is not true

Einstein's theory of special relativity resolved the problem. In special relativity, velocities measured in two different reference frames can no longer be added as Newton did, because one observer disagrees with the time intervals and lengths measured by the other observer. As a result, the wave equation has the same form to all inertial observers, with the same value for the speed of light, *c*. Newton's laws of motion are modified in such a way that in all situations they were originally developed for (e.g., uncharged objects moving at speeds much smaller than the speed of light), the differences are so small as to be practically immeasurable. However, when we look at currents it turns out that these very small differences do matter in

In special relativity, all inertial frames are equivalent – meaning that all laws of physics are the same, as they are in Galilean relativity. However, rather than postulating that time and space are the same ("invariant") for all inertial observers, it is postulated that the speed of light *c* is invariant: it is measured to be the same in all reference frames by all inertial observers. As a consequence, measurements of time and space made in one reference frame that is moving with respect to another are different – even though the measurements may be made in the exact same way as seen from within each reference system. Seen from one reference system, a clock travelling at constant speed appears to be ticking more slowly, and appears contracted in the direction of motion. Also, if there is more than one clock at different locations, the clocks can only be synchronized according to one observer, but not simultaneously to another

These ideas can be investigated with an imaginary device – a light clock. Because both observers agree that light travels at speed *c* in both reference frames, this allows us to compare

Substituting all this back into the wave equation, and grouping judiciously, we obtain

 *<sup>∂</sup><sup>t</sup>* � *∂t*

*∂x*� *∂x ∂t* � *∂x* + *∂*2*E ∂t*�<sup>2</sup>

*∂x*� *∂t ∂t* � *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

> *∂t* � *∂x*

2 

*c*2

*∂*2*E*

 *∂t* � *∂x*

 *∂t* � *∂t*

<sup>=</sup> <sup>2</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�*∂t*�

*∂*2*E ∂t*�<sup>2</sup> 2

2

 1 *c*2 *∂x*� *∂t ∂t* � *<sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>∂</sup>x*� *∂x ∂t* � *∂x* .

*<sup>∂</sup>x*�*∂t*� . (36)

. (33)

. (34)

(35)

 *∂x*� *∂x*

 *∂x*� *∂t*

*∂*2*E <sup>∂</sup>x*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�<sup>2</sup>

*∂*2*E <sup>∂</sup>t*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�<sup>2</sup>

 *∂x*� *∂t*

for the Galilean transformation, since we obtain:

*∂*2*E ∂x*�<sup>2</sup>  <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2 − 1 *c*2 *∂*2*E <sup>∂</sup>t*�<sup>2</sup> <sup>=</sup> <sup>−</sup>2*<sup>v</sup>*

*∂*2*E ∂x*�<sup>2</sup>  *<sup>∂</sup>x*� *∂x*

2 − 1 *c*2

**4.3 Principles of special relativity**

observer in a different reference frame.

everyday situations.

The second derivative with respect to time is, likewise:

2 − 1 *c*2 *∂*2*E ∂t*�<sup>2</sup>

Equations (26) and (27) are known as a Galilean transformation. It is easy to see that if Newton's second law holds for one observer, it automatically holds for the other. For an object moving at speed *u* we find that

$$
\mu' \equiv \frac{\mathbf{dx}'}{\mathbf{dt}'} = \frac{\mathbf{dx}'}{\mathbf{dt}} = \frac{\mathbf{dx}}{\mathbf{dt}} - v = \mu - v,\tag{28}
$$

so we get

$$a' = \frac{\mathbf{d}^2 \mathbf{x}'}{\mathbf{d}t'^2} = \frac{\mathbf{d}^2 \mathbf{x}'}{\mathbf{d}t^2} = \frac{\mathbf{d}^2 \mathbf{x}}{\mathbf{d}t^2} = a. \tag{29}$$

Hence, in both reference frames, the accelerations are the same, and hence the forces are the same, too.

#### **4.2 The wave equation in two inertial reference frames**

A problem occurs when we consider light waves. The transformation (28) implies that, in a rest frame travelling at the speed of light *c* with respect to an emitter, light would be at rest – it is not clear how that could be.

To put this problem on a firmer mathematical footing, we derive the general linear transformation of the wave equation; we then substitute in the Galilean transformation. For an electromagnetic wave, the electric field *E* satisfies, in one reference frame,

$$
\frac{
\partial^2 E}{
\partial x^2} - \frac{1}{c^2} \frac{
\partial^2 E}{
\partial t^2} = 0.
\tag{30}
$$

We can express the derivative with respect to *x* in terms of variables used in another reference frame, *x*� and *t* � , by using the chain rule:

$$
\frac{\partial E}{\partial \mathbf{x}} = \frac{\partial E}{\partial \mathbf{x}'} \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} + \frac{\partial E}{\partial t'} \frac{\partial t'}{\partial \mathbf{x}}.\tag{31}
$$

The second derivative contains five terms:

$$\frac{\partial^2 E}{\partial \mathbf{x}^2} = \frac{\partial^2 E}{\partial \mathbf{x}'^2} \left( \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} \right)^2 + 2 \frac{\partial^2 E}{\partial \mathbf{x}' \partial t'} \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} \frac{\partial t'}{\partial \mathbf{x}} + \frac{\partial^2 \mathbf{x}'}{\partial \mathbf{x}^2} \frac{\partial E}{\partial \mathbf{x}'} + \frac{\partial^2 E}{\partial t'^2} \left( \frac{\partial t'}{\partial \mathbf{x}} \right)^2 + \frac{\partial^2 t'}{\partial \mathbf{x}^2} \frac{\partial E}{\partial t'}.\tag{32}$$

8 Will-be-set-by-IN-TECH

various times *t*. An observer in a second reference frame moves with speed *v* relative to the first frame, with identical, synchronized clocks and metre sticks. Time intervals and lengths

The second observer sees the first observer move away at speed *v*. The distance between

*t*

Equations (26) and (27) are known as a Galilean transformation. It is easy to see that if Newton's second law holds for one observer, it automatically holds for the other. For an

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>d</sup>*<sup>x</sup>*

Hence, in both reference frames, the accelerations are the same, and hence the forces are the

A problem occurs when we consider light waves. The transformation (28) implies that, in a rest frame travelling at the speed of light *c* with respect to an emitter, light would be at rest –

To put this problem on a firmer mathematical footing, we derive the general linear transformation of the wave equation; we then substitute in the Galilean transformation. For

We can express the derivative with respect to *x* in terms of variables used in another reference

*∂x*� *∂x* + *∂E ∂t*� *∂t* �

> *∂E <sup>∂</sup>x*� <sup>+</sup>

*∂*2*E ∂t*�<sup>2</sup>  *∂t* � *∂x*

*∂x*� *∂x ∂t* � *∂x* + *∂*2*x*� *∂x*<sup>2</sup>

<sup>d</sup>*t*<sup>2</sup> <sup>=</sup> d2*<sup>x</sup>*

<sup>d</sup>*t*�<sup>2</sup> <sup>=</sup> d2*x*�

. Hence, the second observer can use the

� = *t* (27)

<sup>d</sup>*<sup>t</sup>* <sup>−</sup> *<sup>v</sup>* <sup>=</sup> *<sup>u</sup>* <sup>−</sup> *<sup>v</sup>*, (28)

<sup>d</sup>*t*<sup>2</sup> <sup>=</sup> *<sup>a</sup>*. (29)

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> 0. (30)

*<sup>∂</sup><sup>x</sup>* . (31)

*∂E*

*<sup>∂</sup>t*� . (32)

2 + *∂*2*t* � *∂x*<sup>2</sup>

*x*� = *x* − *vt* (26)

� is given by *vt*�

measurements of the first observer, provided the following changes are made:

<sup>d</sup>*t*� <sup>=</sup> <sup>d</sup>*x*�

*<sup>u</sup>*� <sup>≡</sup> <sup>d</sup>*x*�

**4.2 The wave equation in two inertial reference frames**

, by using the chain rule:

2

*<sup>a</sup>*� <sup>=</sup> d2*x*�

an electromagnetic wave, the electric field *E* satisfies, in one reference frame,

*∂E <sup>∂</sup><sup>x</sup>* <sup>=</sup> *<sup>∂</sup><sup>E</sup> ∂x*�

<sup>+</sup> <sup>2</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�*∂t*�

*∂*2*E <sup>∂</sup>x*<sup>2</sup> <sup>−</sup> <sup>1</sup> *c*2 *∂*2*E*

are assumed to be same for both observers.

the two observers at a time *t*

object moving at speed *u* we find that

it is not clear how that could be.

�

*∂*2*E <sup>∂</sup>x*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*<sup>E</sup> ∂x*�<sup>2</sup>

The second derivative contains five terms:

 *∂x*� *∂x*

so we get

same, too.

frame, *x*� and *t*

For linear transformations, the third and fifth terms are zero. Hence we obtain:

$$\frac{\partial^2 E}{\partial \mathbf{x}^2} = \frac{\partial^2 E}{\partial \mathbf{x}'^2} \left( \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} \right)^2 + 2 \frac{\partial^2 E}{\partial \mathbf{x}' \partial t'} \frac{\partial \mathbf{x}'}{\partial \mathbf{x}} \frac{\partial t'}{\partial \mathbf{x}} + \frac{\partial^2 E}{\partial t'^2} \left( \frac{\partial t'}{\partial \mathbf{x}} \right)^2. \tag{33}$$

The second derivative with respect to time is, likewise:

$$\frac{\partial^2 E}{\partial t^2} = \frac{\partial^2 E}{\partial \mathbf{x}'^2} \left( \frac{\partial \mathbf{x}'}{\partial t} \right)^2 + 2 \frac{\partial^2 E}{\partial \mathbf{x}' \partial t'} \frac{\partial \mathbf{x}'}{\partial t} \frac{\partial t'}{\partial t} + \frac{\partial^2 E}{\partial t'^2} \left( \frac{\partial t'}{\partial t} \right)^2. \tag{34}$$

Substituting all this back into the wave equation, and grouping judiciously, we obtain

$$2\frac{\partial^2 E}{\partial \mathbf{x'}^2} \left[ \left( \frac{\partial \mathbf{x'}}{\partial \mathbf{x}} \right)^2 - \frac{1}{c^2} \left( \frac{\partial \mathbf{x'}}{\partial t} \right)^2 \right] - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} \left[ \left( \frac{\partial t'}{\partial t} \right)^2 - c^2 \left( \frac{\partial t'}{\partial \mathbf{x}} \right)^2 \right] = 2 \frac{\partial^2 E}{\partial \mathbf{x'} \partial t'} \left[ \frac{1}{c^2} \frac{\partial \mathbf{x'}}{\partial t} \frac{\partial t'}{\partial t} - \frac{\partial \mathbf{x'}}{\partial \mathbf{x}} \frac{\partial t'}{\partial \mathbf{x}} \right] \tag{35}$$

To retain the wave equation (30), it is clear that the right-hand side of this equation must be zero while the terms in square brackets on the left-hand side must be equal. This is not true for the Galilean transformation, since we obtain:

$$\frac{\partial^2 E}{\partial \mathbf{x}'^2} \left( 1 - \frac{v^2}{c^2} \right) - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = -\frac{2v}{c^2} \frac{\partial^2 E}{\partial \mathbf{x}' \partial t'}.\tag{36}$$

#### **4.3 Principles of special relativity**

Einstein's theory of special relativity resolved the problem. In special relativity, velocities measured in two different reference frames can no longer be added as Newton did, because one observer disagrees with the time intervals and lengths measured by the other observer. As a result, the wave equation has the same form to all inertial observers, with the same value for the speed of light, *c*. Newton's laws of motion are modified in such a way that in all situations they were originally developed for (e.g., uncharged objects moving at speeds much smaller than the speed of light), the differences are so small as to be practically immeasurable. However, when we look at currents it turns out that these very small differences do matter in everyday situations.

In special relativity, all inertial frames are equivalent – meaning that all laws of physics are the same, as they are in Galilean relativity. However, rather than postulating that time and space are the same ("invariant") for all inertial observers, it is postulated that the speed of light *c* is invariant: it is measured to be the same in all reference frames by all inertial observers. As a consequence, measurements of time and space made in one reference frame that is moving with respect to another are different – even though the measurements may be made in the exact same way as seen from within each reference system. Seen from one reference system, a clock travelling at constant speed appears to be ticking more slowly, and appears contracted in the direction of motion. Also, if there is more than one clock at different locations, the clocks can only be synchronized according to one observer, but not simultaneously to another observer in a different reference frame.

These ideas can be investigated with an imaginary device – a light clock. Because both observers agree that light travels at speed *c* in both reference frames, this allows us to compare .

measurements in the two reference frames. Both observers agree that their own light clock consists of two mirrors mounted on a ruler a distance *l*<sup>0</sup> apart, and that it takes a light pulse a time interval Δ*t*<sup>0</sup> for a round trip. Both observers agree that *l*<sup>0</sup> and Δ*t*<sup>0</sup> are related by

$$d\_0 = \mathbf{c} \cdot \frac{1}{2} \Delta t\_{0\prime} \tag{37}$$

*v*Δ*t LR*

*l=l0 /*γ

variables are defined in the text.

Now, defining

*c*

*v*Δ*t*/2

*v*Δ*t*

*c*

Current-Carrying Wires and Special Relativity 13

*v v v*

*c c c*

Fig. 4. A light clock in frame 2 as measured by observer 1, when the light clock is (a) perpendicular and (b) parallel to the relative motion of the two frames at speed *v*. The dot indicates a photon travelling at speed *c* in the direction indicated by the arrow. Other

This reasoning can be quantified using Pythagoras' Theorem. Observer 1 sees that

The time interval <sup>Δ</sup>*<sup>t</sup>* can be related to the time on clock 1, <sup>Δ</sup>*t*, because *<sup>l</sup>*<sup>0</sup> <sup>=</sup> *<sup>c</sup>* · <sup>1</sup>

*<sup>γ</sup>* <sup>≡</sup> <sup>1</sup> <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2

Since all processes in frame 2 are in sync with clock 2, observer 1 sees all processes in frame 2 run slower than those in frame 1 by a factor *γ*. Conversely, to observer 2, everything is normal

Now both observers turn their clocks through 90 degrees, so the light travels parallel to their relative motion, as shown in Figure 4b. Within their own reference frames, the clocks still run

in frame 2; but observer 2 sees all processes in frame 1 run slow by the same factor *γ*.

Δ*t* 2 <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2 = Δ*t* 2

substituting, taking the square root and dividing by *γ*, we conclude that

**4.6 Lengths parallel to relative motion are contracted**

 *c* · 1 2 Δ*t* <sup>2</sup> − *v* · 1 2 Δ*t* <sup>2</sup> = *l* 2 (a)

<sup>0</sup>. (38)

<sup>0</sup>. (39)

, (40)

Δ*t* = *γ*Δ*t*0. (41)

<sup>2</sup>Δ*t*0; hence

*l=l0*

*c*

*v*Δ*t* (b)

*v v v*

and both agree that this is true irrespective of the orientation of the light clock.

However, when comparing each other's measurements, the observers are in for some surprises. As motion in one direction is independent from motion in an orthogonal direction, it makes sense to distinguish between lengths parallel and perpendicular to the relative motion of the two reference frames. A very useful sequence of looking at the light clock was given by Mermin (1989):


#### **4.4 Lengths perpendicular to motion are unaffected**

In the first thought experiments, each observer has a light clock. They are parallel to each other, and perpendicular to their relative motion (see Figure 4a). We can imagine that a piece of chalk is attached to each end of each clock, so that when the two clocks overlap, each makes a mark on the other.

We arrive at a result by reductio ad absurdum. Say that observer 1 sees clock 2 contract (but his own does not, of course – the observed contraction would be due purely to relative motion). Both observers would agree on the marks made by the pieces of chalk on clock 2 – they are inside the ends of clock 1. They would also both agree that the ends of clock 1 do not mark clock 2. Special relativity demands that the laws of physics are the same for both observers: so observer 2 must see clock 1 shrink by the same factor, clock 2 retains the same length; and hence the chalk marks on clock 2 are inside the ends, while there are no marks on clock 1. Thus, we arrive at a contradiction. Assuming one observer sees the other's clocks expand lead to the same conundrum. The only possible conclusion: both observers agree that both clocks have length *l*<sup>0</sup> in both frames.

#### **4.5 Time intervals: moving clocks run more slowly**

In the same set-up, observer 1 sees the light pulse in his clock move vertically, while the light pulse in clock 2 moves diagonally (see Figure 4a). Observer 1 uses his measurements only, plus the information that clock 2 has length *l*<sup>0</sup> and that the light pulse of clock 2 moves at speed *c*, also as measured by observer 1. Observer 1 measures that a pulse in clock 2 goes from the bottom mirror to the top and back again in an interval Δ*t*, which must be greater than Δ*t*0, as the light bouncing between the mirrors travels further at the same speed. Thus, as seen by observer 1, clock 2 takes longer to complete a tick, and runs slow; clock 1 has already started a second cycle when clock 2 completes its first.

10 Will-be-set-by-IN-TECH

measurements in the two reference frames. Both observers agree that their own light clock consists of two mirrors mounted on a ruler a distance *l*<sup>0</sup> apart, and that it takes a light pulse a

> 1 2

However, when comparing each other's measurements, the observers are in for some surprises. As motion in one direction is independent from motion in an orthogonal direction, it makes sense to distinguish between lengths parallel and perpendicular to the relative motion of the two reference frames. A very useful sequence of looking at the light clock was

In the first thought experiments, each observer has a light clock. They are parallel to each other, and perpendicular to their relative motion (see Figure 4a). We can imagine that a piece of chalk is attached to each end of each clock, so that when the two clocks overlap, each makes

We arrive at a result by reductio ad absurdum. Say that observer 1 sees clock 2 contract (but his own does not, of course – the observed contraction would be due purely to relative motion). Both observers would agree on the marks made by the pieces of chalk on clock 2 – they are inside the ends of clock 1. They would also both agree that the ends of clock 1 do not mark clock 2. Special relativity demands that the laws of physics are the same for both observers: so observer 2 must see clock 1 shrink by the same factor, clock 2 retains the same length; and hence the chalk marks on clock 2 are inside the ends, while there are no marks on clock 1. Thus, we arrive at a contradiction. Assuming one observer sees the other's clocks expand lead to the same conundrum. The only possible conclusion: both observers agree that both

In the same set-up, observer 1 sees the light pulse in his clock move vertically, while the light pulse in clock 2 moves diagonally (see Figure 4a). Observer 1 uses his measurements only, plus the information that clock 2 has length *l*<sup>0</sup> and that the light pulse of clock 2 moves at speed *c*, also as measured by observer 1. Observer 1 measures that a pulse in clock 2 goes from the bottom mirror to the top and back again in an interval Δ*t*, which must be greater than Δ*t*0, as the light bouncing between the mirrors travels further at the same speed. Thus, as seen by observer 1, clock 2 takes longer to complete a tick, and runs slow; clock 1 has

Δ*t*0, (37)

time interval Δ*t*<sup>0</sup> for a round trip. Both observers agree that *l*<sup>0</sup> and Δ*t*<sup>0</sup> are related by

*l*<sup>0</sup> = *c* ·

and both agree that this is true irrespective of the orientation of the light clock.

given by Mermin (1989):

3. length parallel to motion 4. synchronization of clocks

2. time intervals

a mark on the other.

1. length perpendicular to motion

clocks have length *l*<sup>0</sup> in both frames.

**4.5 Time intervals: moving clocks run more slowly**

already started a second cycle when clock 2 completes its first.

**4.4 Lengths perpendicular to motion are unaffected**

Fig. 4. A light clock in frame 2 as measured by observer 1, when the light clock is (a) perpendicular and (b) parallel to the relative motion of the two frames at speed *v*. The dot indicates a photon travelling at speed *c* in the direction indicated by the arrow. Other variables are defined in the text.

This reasoning can be quantified using Pythagoras' Theorem. Observer 1 sees that

$$\left(c \cdot \frac{1}{2}\Delta t\right)^2 - \left(v \cdot \frac{1}{2}\Delta t\right)^2 = l\_0^2. \tag{38}$$

The time interval <sup>Δ</sup>*<sup>t</sup>* can be related to the time on clock 1, <sup>Δ</sup>*t*, because *<sup>l</sup>*<sup>0</sup> <sup>=</sup> *<sup>c</sup>* · <sup>1</sup> <sup>2</sup>Δ*t*0; hence

$$
\Delta t^2 \left( 1 - \frac{v^2}{c^2} \right) = \Delta t\_0^2. \tag{39}
$$

Now, defining

$$\gamma \equiv \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \,\,\,\tag{40}$$

substituting, taking the square root and dividing by *γ*, we conclude that

$$
\Delta t = \gamma \Delta t\_0. \tag{41}
$$

Since all processes in frame 2 are in sync with clock 2, observer 1 sees all processes in frame 2 run slower than those in frame 1 by a factor *γ*. Conversely, to observer 2, everything is normal in frame 2; but observer 2 sees all processes in frame 1 run slow by the same factor *γ*.

#### **4.6 Lengths parallel to relative motion are contracted**

Now both observers turn their clocks through 90 degrees, so the light travels parallel to their relative motion, as shown in Figure 4b. Within their own reference frames, the clocks still run

However, Newton's Second Law does not transform in special relativity. In the situations under discussion in Section 5.1, all forces are perpendicular to the relative speed *v*. In that case, a force of magnitude *F*<sup>0</sup> in the rest frame is measured by an observer in a moving frame

Current-Carrying Wires and Special Relativity 15

An operational definition for a transverse force is given by Martins (1982). For the sake of

The considerations of the three previous sections can be brought together quite neatly. We model a current-carrying wire as a rigid lattice of ions, and a fluid of electrons that are free to move through the lattice. In the reference frame of the ions, then, the electrons move with a certain drift speed, *vd*. But, by the same token, in the frame of the electrons, the ions move

Experimental evidence shows that a stationary charge is not affected by the presence of a current-carrying wire. This absence of a net electrostatic force implies that the ion and electron charge densities in a current-carrying wire must have the same magnitude. This statement is

Consider the case of zero current. Call the linear charge density of the ions *λ*0. By charge neutrality, the linear charge density of the electrons must be equal to −*λ*0. Now let the electrons move at drift speed *vd* relative to the ions, causing a current *I*. Experimentally, both linear charge densities remain unchanged, since a stationary charged object placed near the wire does not experience a net force. So, as seen in the ion frame, the linear electron charge

Moreover, in the electron frame, the ions are moving, and hence their linear charge density is

completeness we note that a parallel force transforms as *F* = *F*0/*γ*3.

3. An charged object moving parallel to a current-carrying wire at speed *vd*;

In the electron frame, the linear charge density of the electrons must be

*λ*<sup>−</sup> = *γλ*�

*λ*�

*λ*�

**5. Electric fields, magnetic fields, and special relativity**

*F* = *F*0/*γ*. (47)

*λ*<sup>−</sup> = −*λ*0. (48)

<sup>−</sup> = −*λ*0/*γ*, (49)

<sup>−</sup> = *<sup>γ</sup>* · (−*λ*0/*γ*) = −*λ*0. (50)

<sup>+</sup> = *γλ*0. (51)

to have magnitude *F* given by

with a speed *vd*.

density is given by:

so that

We will consider four cases:

1. An infinitely thin current-carrying wire; 2. A current-carrying wire of finite width;

4. Two parallel current-carrying wires.

**5.1 Length contraction in a current carrying wire**

more problematic than it may seem at first glance.

at the same rate; hence each observer sees the other's clock run slow by a factor *γ* as before.

However, to observer 1, after the light pulse leaves the left mirror of clock 2, that whole clock travels to the right. The pulse thus travels by a distance *l* + *v* · Δ*tLR* at speed *c* during a time interval Δ*tLR* before it hits the right mirror, where *l* is the length of the clock as measured by observer 1. Hence

$$d + v \cdot \Delta t\_{LR} = c \cdot \Delta t\_{LR}.\tag{42}$$

Similarly, after the pulse reflects it travels a distance *l* − *v* · Δ*tRL* before it hits the left mirror again. Observer 1 finds for the total time Δ*t*

$$
\Delta t = \Delta t\_{LR} + \Delta t\_{RL} = \frac{l}{c - v} + \frac{l}{c + v} = \frac{2l}{c} \cdot \frac{1}{1 - \frac{v^2}{c^2}} = \gamma^2 \frac{2l}{c}.\tag{43}
$$

The time interval Δ*t* can be linked to the time interval in frame 1, Δ*t*0, by (41), which, in turn, is linked to the length in frame 1, Δ*l*0, by (37). Straightforward substitution yields

$$l = \frac{l\_0}{\gamma}.\tag{44}$$

Thus, as measured by observer 1, all lengths in frame 2 parallel to the motion are shorter than in frame 1 by a factor *γ* (but both perpendicular lengths are the same). As seen by observer 2, everything is normal in frame 2, but all parallel lengths in frame 1 are contracted by the same factor *γ*. When the two observers investigate each other's metre sticks, they both agree on how many atoms there are in the each stick, but disagree on the spacing between them.

#### **4.7 Synchronization of clocks is only possible in one frame at a time**

As it stands, it is hard to see how the observations in both frames can be reconciled. How can *both* observers see the other clocks run slowly, and the other's lengths contracted? The answer lies in synchronization. Without going into much detail, we outline some key points here.

Measuring the length of an object requires, in principle, the determination of two locations (the ends of the object) at the same time. However, when two clocks are synchronized in frame 1 according to observer 1, they are not according to observer 2. As the frames move with respect to each other, observer 2 concludes that observer 1 moved his ruler while determining the position of each end of the object. In the end, each observer can explain all measurements in a consistent fashion. For an accessible yet rigorous in-depth discussion see Mermin (1989). The end result is the transformation laws

$$\mathbf{x}' = \gamma(\mathbf{x} - vt) \tag{45}$$

$$t' = \gamma(t - v\mathbf{x}/c^2) \tag{46}$$

#### **4.8 Transformation of forces and invariance of the wave equation in special relativity**

Substituting the transformations of special relativity into the wave equation (35) shows that the wave equation has the same form in both frames: the two factors in square brackets are equal to 1, and the right hand side is equal to zero.

12 Will-be-set-by-IN-TECH

at the same rate; hence each observer sees the other's clock run slow by a factor *γ* as before. However, to observer 1, after the light pulse leaves the left mirror of clock 2, that whole clock travels to the right. The pulse thus travels by a distance *l* + *v* · Δ*tLR* at speed *c* during a time interval Δ*tLR* before it hits the right mirror, where *l* is the length of the clock as measured by

Similarly, after the pulse reflects it travels a distance *l* − *v* · Δ*tRL* before it hits the left mirror

The time interval Δ*t* can be linked to the time interval in frame 1, Δ*t*0, by (41), which, in turn,

*<sup>l</sup>* <sup>=</sup> *<sup>l</sup>*<sup>0</sup>

Thus, as measured by observer 1, all lengths in frame 2 parallel to the motion are shorter than in frame 1 by a factor *γ* (but both perpendicular lengths are the same). As seen by observer 2, everything is normal in frame 2, but all parallel lengths in frame 1 are contracted by the same factor *γ*. When the two observers investigate each other's metre sticks, they both agree on how many atoms there are in the each stick, but disagree on the spacing between them.

As it stands, it is hard to see how the observations in both frames can be reconciled. How can *both* observers see the other clocks run slowly, and the other's lengths contracted? The answer lies in synchronization. Without going into much detail, we outline some key points here.

Measuring the length of an object requires, in principle, the determination of two locations (the ends of the object) at the same time. However, when two clocks are synchronized in frame 1 according to observer 1, they are not according to observer 2. As the frames move with respect to each other, observer 2 concludes that observer 1 moved his ruler while determining the position of each end of the object. In the end, each observer can explain all measurements in a consistent fashion. For an accessible yet rigorous in-depth discussion see Mermin (1989).

*c* − *v* + *l <sup>c</sup>* <sup>+</sup> *<sup>v</sup>* <sup>=</sup> <sup>2</sup>*<sup>l</sup>*

is linked to the length in frame 1, Δ*l*0, by (37). Straightforward substitution yields

**4.7 Synchronization of clocks is only possible in one frame at a time**

*t*

**4.8 Transformation of forces and invariance of the wave equation in special relativity**

Substituting the transformations of special relativity into the wave equation (35) shows that the wave equation has the same form in both frames: the two factors in square brackets are

*l* + *v* · Δ*tLR* = *c* · Δ*tLR*. (42)

<sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>2</sup>*<sup>l</sup>*

*<sup>γ</sup>* . (44)

*x*� = *γ*(*x* − *vt*) (45)

� <sup>=</sup> *<sup>γ</sup>*(*<sup>t</sup>* <sup>−</sup> *vx*/*c*2) (46)

*<sup>c</sup>* . (43)

*<sup>c</sup>* · <sup>1</sup> <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2

observer 1. Hence

again. Observer 1 finds for the total time Δ*t*

The end result is the transformation laws

equal to 1, and the right hand side is equal to zero.

<sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> <sup>Δ</sup>*tLR* <sup>+</sup> <sup>Δ</sup>*tRL* <sup>=</sup> *<sup>l</sup>*

However, Newton's Second Law does not transform in special relativity. In the situations under discussion in Section 5.1, all forces are perpendicular to the relative speed *v*. In that case, a force of magnitude *F*<sup>0</sup> in the rest frame is measured by an observer in a moving frame to have magnitude *F* given by

$$F = F\_0 / \gamma. \tag{47}$$

An operational definition for a transverse force is given by Martins (1982). For the sake of completeness we note that a parallel force transforms as *F* = *F*0/*γ*3.

## **5. Electric fields, magnetic fields, and special relativity**

The considerations of the three previous sections can be brought together quite neatly. We model a current-carrying wire as a rigid lattice of ions, and a fluid of electrons that are free to move through the lattice. In the reference frame of the ions, then, the electrons move with a certain drift speed, *vd*. But, by the same token, in the frame of the electrons, the ions move with a speed *vd*.

We will consider four cases:


#### **5.1 Length contraction in a current carrying wire**

Experimental evidence shows that a stationary charge is not affected by the presence of a current-carrying wire. This absence of a net electrostatic force implies that the ion and electron charge densities in a current-carrying wire must have the same magnitude. This statement is more problematic than it may seem at first glance.

Consider the case of zero current. Call the linear charge density of the ions *λ*0. By charge neutrality, the linear charge density of the electrons must be equal to −*λ*0. Now let the electrons move at drift speed *vd* relative to the ions, causing a current *I*. Experimentally, both linear charge densities remain unchanged, since a stationary charged object placed near the wire does not experience a net force. So, as seen in the ion frame, the linear electron charge density is given by:

$$
\lambda\_- = -\lambda\_0. \tag{48}
$$

In the electron frame, the linear charge density of the electrons must be

$$
\lambda'\_- = -\lambda\_0 / \gamma\_\prime \tag{49}
$$

so that

$$
\lambda\_- = \gamma \lambda\_-^{\prime} = \gamma \cdot (-\lambda\_0/\gamma) = -\lambda\_0. \tag{50}
$$

Moreover, in the electron frame, the ions are moving, and hence their linear charge density is

$$
\lambda'\_+ = \gamma \lambda\_0. \tag{51}
$$

Combining the two yields

can be approximated as a surface density:

a "self-pinched" uniform current density given by

and charge were located on the central axis of the wire.

*R*/γ *R*

**5.3 A charged object near a current carrying wire**

*I*/π*R*<sup>2</sup>

*J*=0

*J* = γ<sup>2</sup>

ion frame.

*a* = *R*/*γ*. (57)

<sup>+</sup> · 2*πR* = 2*γλ*0/*R*. (58)

*<sup>π</sup>R*<sup>2</sup> (59)

λ=γλ<sup>0</sup> *v*2 d

<sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>d</sup>*/*c*2, where *vd*

/*c*<sup>2</sup> <sup>ρ</sup>=0

/π*R*<sup>2</sup>

ρ=γλ<sup>0</sup>

ρ=0

*R*/γ *R*

Thus, the wire is electrically neutral between 0 and *R*/*γ*, and has a positive volume charge density given by (55) between *R*/*γ* and *R*. Because in practice the outer shell is very thin, it

Current-Carrying Wires and Special Relativity 17

As we have seen, lengths perpendicular to the motion do not change. Hence in the ion frame the electrons comprise a uniform line of electrons moving at speed *vd*; in other words, there is

*<sup>π</sup>a*<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> *<sup>I</sup>*

Figure 5 shows some relevant current and volume charge densities in both reference frames. Note that, by the considerations of Section 2.4, for *r* > *R* we may treat the wire as if all current

ion frame electron frame

*<sup>J</sup>* <sup>=</sup> *<sup>I</sup>*

*σ*� = *ρ*�

between 0 and *R*/*γ*, and zero current density between *R*/*γ* and *R*.

Fig. 5. Current and charge densities in a current-carrying wire. *<sup>γ</sup>* <sup>=</sup> 1/

is the relative speed of the ions and electrons, and *λ*<sup>0</sup> is the linear ion density as seen in the

We have established that in the ion frame, a current-carrying wire does not exert an electrostatic force; but in the electron frame, it does. There is nothing wrong with this, but we

The net charge density in the electron frame, *λ*� , is then given by

$$
\lambda' = \lambda\_+' + \lambda\_-' = \gamma \lambda\_0 - \lambda\_0/\gamma = \lambda\_0 \gamma \left(1 - 1/\gamma^2\right) = \lambda\_0 \gamma v\_d^2/c^2. \tag{52}
$$

Thus, in the electron frame, the wire is charged. We cannot, however, simply assume that Coulomb's Law (6) holds; that law was obtained from experiments on stationary charges, while the ions are moving in the electron frame. In fact, the magnitude of the electric field d*E* due to a point charge *λ*d*z* moving at speed *vd* is given by (French, 1968; Purcell, 1984)

$$\mathrm{d}E = \frac{\lambda \mathrm{d}z \left(1 - v\_d^2/c^2\right)}{4\pi\epsilon\_0(r^2 + z^2)\left(1 - \frac{v\_d^2}{c^2}\frac{r^2}{r^2 + z^2}\right)},\tag{53}$$

using the notation of Figure 1a. However, when we integrate the radial component of this electric field, we *do* obtain the same result; switching to primed coordinates to denote the electron frame,

$$E' = \frac{\lambda'}{2\pi\epsilon\_0 r'} = \frac{\lambda\_0 \gamma v\_d^2}{2\pi\epsilon\_0 r}.\tag{54}$$

We have used the fact that lengths perpendicular to motion do not contract; hence *r* = *r*� .

#### **5.2 Current and charge distribution within a wire**

Now consider a wire of finite radius, *R*. We can model this as an infinite number of parallel infinitely thin wires placed in a circle. Assume that each wire starts out as discussed above.

As seen in the ion frame, there are many electron currents in the same direction; each current will set up a magnetostatic field, the net effect of which will be an attraction towards the centre. However, once the electrons start to migrate towards the centre, a net negative charge is created in the centre of the wire; equilibrium is established when the two cancel (Gabuzda, 1993; Matzek & Russell, 1968).

As seen in the electron frame, there is a current of positive ions, but since the ion frame is assumed perfectly rigid, no redistribution of charge occurs as a result. However, due to length contraction there is also a net positive charge density, which will attract the electrons towards the centre of the wire (Gabuzda, 1993). This must occur in such a way that the net electric field is zero; this, in turn, can only happen if the net volume charge density is zero. Consequently, the linear electron density is distorted: within a radius *a*, a uniform electron volume charge density is established that is equal to the ion volume charge density; between *a* and *R*, the electron density is zero.

The magnitude of this effect can be calculated easily. The uniform ion volume charge density is given by

$$
\rho\_{+}^{\prime} = \frac{\lambda\_{+}^{\prime}}{\pi \mathcal{R}^2} = \frac{\gamma \lambda\_0}{\pi \mathcal{R}^2} ; \tag{55}
$$

this must be equal to (minus) the uniform electron volume density over a radius *a*. Hence we obtain

$$
\rho\_{-}^{\prime} = \frac{\lambda\_{-}^{\prime}}{\pi a^2} = -\frac{\lambda\_0}{\gamma \pi a^2}.\tag{56}
$$

Combining the two yields

14 Will-be-set-by-IN-TECH

Thus, in the electron frame, the wire is charged. We cannot, however, simply assume that Coulomb's Law (6) holds; that law was obtained from experiments on stationary charges, while the ions are moving in the electron frame. In fact, the magnitude of the electric field d*E*

> <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

using the notation of Figure 1a. However, when we integrate the radial component of this electric field, we *do* obtain the same result; switching to primed coordinates to denote the

<sup>2</sup>*π�*0*r*� <sup>=</sup> *<sup>λ</sup>*0*γv*<sup>2</sup>

 <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *d c*2 *r*2 *r*<sup>2</sup>+*z*<sup>2</sup>

<sup>−</sup> = *γλ*<sup>0</sup> − *<sup>λ</sup>*0/*<sup>γ</sup>* = *<sup>λ</sup>*0*<sup>γ</sup>*

due to a point charge *λ*d*z* moving at speed *vd* is given by (French, 1968; Purcell, 1984)

4*π�*0(*r*<sup>2</sup> + *z*2)

*<sup>E</sup>*� <sup>=</sup> *<sup>λ</sup>*�

We have used the fact that lengths perpendicular to motion do not contract; hence *r* = *r*�

Now consider a wire of finite radius, *R*. We can model this as an infinite number of parallel infinitely thin wires placed in a circle. Assume that each wire starts out as discussed above. As seen in the ion frame, there are many electron currents in the same direction; each current will set up a magnetostatic field, the net effect of which will be an attraction towards the centre. However, once the electrons start to migrate towards the centre, a net negative charge is created in the centre of the wire; equilibrium is established when the two cancel (Gabuzda,

As seen in the electron frame, there is a current of positive ions, but since the ion frame is assumed perfectly rigid, no redistribution of charge occurs as a result. However, due to length contraction there is also a net positive charge density, which will attract the electrons towards the centre of the wire (Gabuzda, 1993). This must occur in such a way that the net electric field is zero; this, in turn, can only happen if the net volume charge density is zero. Consequently, the linear electron density is distorted: within a radius *a*, a uniform electron volume charge density is established that is equal to the ion volume charge density; between *a* and *R*, the

The magnitude of this effect can be calculated easily. The uniform ion volume charge density

this must be equal to (minus) the uniform electron volume density over a radius *a*. Hence we

*ρ*� <sup>+</sup> <sup>=</sup> *<sup>λ</sup>*� + *<sup>π</sup>R*<sup>2</sup> <sup>=</sup> *γλ*<sup>0</sup>

*ρ*� <sup>−</sup> <sup>=</sup> *<sup>λ</sup>*� − *<sup>π</sup>a*<sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>λ</sup>*<sup>0</sup>

<sup>d</sup>*<sup>E</sup>* <sup>=</sup> *<sup>λ</sup>*d*<sup>z</sup>*

, is then given by

<sup>1</sup> <sup>−</sup> 1/*γ*<sup>2</sup>

= *λ*0*γv*<sup>2</sup>

*<sup>d</sup>*/*c*2. (52)

, (53)

. (54)

*<sup>π</sup>R*<sup>2</sup> ; (55)

*γπa*<sup>2</sup> . (56)

.

*d*/*c*<sup>2</sup>

*d* 2*π�*0*r*

The net charge density in the electron frame, *λ*�

<sup>+</sup> + *λ*�

**5.2 Current and charge distribution within a wire**

1993; Matzek & Russell, 1968).

electron density is zero.

is given by

obtain

*λ*� = *λ*�

electron frame,

$$a = \mathbb{R} / \gamma. \tag{57}$$

Thus, the wire is electrically neutral between 0 and *R*/*γ*, and has a positive volume charge density given by (55) between *R*/*γ* and *R*. Because in practice the outer shell is very thin, it can be approximated as a surface density:

$$
\sigma' = \rho'\_+ \cdot 2\pi \text{R} = 2\gamma\lambda\_0/\text{R}.\tag{58}
$$

As we have seen, lengths perpendicular to the motion do not change. Hence in the ion frame the electrons comprise a uniform line of electrons moving at speed *vd*; in other words, there is a "self-pinched" uniform current density given by

$$J = \frac{I}{\pi a^2} = \frac{\gamma^2 I}{\pi R^2} \tag{59}$$

between 0 and *R*/*γ*, and zero current density between *R*/*γ* and *R*.

Figure 5 shows some relevant current and volume charge densities in both reference frames. Note that, by the considerations of Section 2.4, for *r* > *R* we may treat the wire as if all current and charge were located on the central axis of the wire.

Fig. 5. Current and charge densities in a current-carrying wire. *<sup>γ</sup>* <sup>=</sup> 1/ <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *<sup>d</sup>*/*c*2, where *vd* is the relative speed of the ions and electrons, and *λ*<sup>0</sup> is the linear ion density as seen in the ion frame.

#### **5.3 A charged object near a current carrying wire**

We have established that in the ion frame, a current-carrying wire does not exert an electrostatic force; but in the electron frame, it does. There is nothing wrong with this, but we

One convenient way of looking at the problem is by considering a segment of wire 2 of length *l*, as measured in the ion frame. This consists of a segment of ions of length *l*/*γ* as seen in the electron frame, and a segment of electrons of length *γl* as seen in the electron frame. The

Current-Carrying Wires and Special Relativity 19

In the ion frame, we can find the total force on the ion segment by dividing it up into point-like segments of charge density *λ*0, and integrating over the entire length *l*. An identical procedure holds for the electrons. In the electron frame, we integrate ion segments of charge density *γλ*<sup>0</sup> over a length *l*/*γ*, and electron segments of charge density *λ*0/*γ* over a length *γl*. The net result is that all expressions found in the previous paragraph hold, if we replace *q* with *λ*0*l*:

*<sup>m</sup>*<sup>−</sup> = <sup>0</sup> (65)

(66)

(67)

*Fe*<sup>+</sup> = *Fm*<sup>+</sup> = *Fe*<sup>−</sup> = *F*�

*<sup>e</sup>*<sup>−</sup> = *<sup>F</sup>*�

*<sup>m</sup>*<sup>+</sup> <sup>=</sup> *γλ*<sup>2</sup>

*Fm*<sup>−</sup> <sup>=</sup> *<sup>λ</sup>*<sup>2</sup>

In this chapter, we have outlined how electrostatics, magnetostatics and special relativity give consistent results for a few cases involving infinitely long current carrying wires. We have used an online integrator to obtain expression for the electrostatic field due to a hollow uniformly charged cylinder, and derived expressions for a solid uniformly charged cylinder and current-carrying wires from it. We have also derived expressions for self-pinching in a

The author gratefully acknowledges fruitful discussions with Enda McGlynn and Mossy

Arons, A.B. (1996). *Teaching Introductory Physics*, 167–187, John Wiley and Sons, ISBN

French, A.P. (1968). *Special Relativity*, 221–225, 251–254, 262–264, Nelson, ISBN 0-393-09793-5,

Gabuzda, D.C. (1987). Magnetic force due to a current-carrying wire: a paradox and its

Gabuzda, D.C. (1993). The charge densities in a current-carrying wire. Am. J. Phys. 61,

Martins, R. de A. (1982). Force measurement and force transformation in special relativity.

Mathematica online integrator, http://integrals.wolfram.com/; last accessed 30 August 2011. Matzek, M.A. & Russell, B.R. (1968). On the Transverse Electric Field within a Conductor

current-carrying wire, by a factor *γ*, and the creation of a surface charge density.

0*v*2 *dl* 2*π�*0*c*2*r*

0*v*2 *dl* 2*π�*0*c*2*r*

transformation leaves the total charge unaltered: it is *λ*0*l*.

*F*� *<sup>e</sup>*<sup>+</sup> = *F*�

**6. Conclusion**

**7. Acknowledgement**

London.

360–362.

0-471-13707-3, New York.

Am. J. Phys. 50, 1008–1011.

resolution. Am. J. Phys. 55, 420–422.

Carrying a Steady Current. Am. J. Phys. 36, 905–907.

Kelly.

**8. References**

must make sure that the effect on charges near the wire is the same in both frames; otherwise, the principle of relativity would be violated.

First, consider a point-like object of charge *q* stationary in the ion frame. There is no electrostatic force on the object, since there is no net charge; nor is there a magnetostatic force, because the speed of the object is zero. In the electron frame, the electrostatic force *F*� *<sup>e</sup>* is given by

$$F'\_{\varepsilon} = qE' = \frac{q\lambda\_0 \gamma v\_d^2}{2\pi\epsilon\_0 c^2 r'}.\tag{60}$$

There is also a magnetostatic force, *F*� *<sup>m</sup>*, as in the electron frame the object is moving with speed *vd* parallel to a current of positive ions; hence

$$F'\_m = qv\_d \mathcal{B}' = \frac{qv\_d \mu\_0 I}{2\pi r'}.\tag{61}$$

The two forces are readily shown to be equal, as *I* = *λ*� <sup>+</sup>*vd* = *γλ*0*vd* and *<sup>μ</sup>*<sup>0</sup> = 1/*�*0*c*2. The forces cancel, because a current of positive ions attracts a positively charged object while the positive charge density repels it.

As a second case, a point-like object of charge *q*, moving parallel to a current carrying wire at speed *vd* in the ion frame, experiences a purely magnetostatic force due to the electrons in the wire, given by:

$$F = qv\_d \mathcal{B} = \frac{qv\_d \mu\_0 I}{2\pi r} = \frac{qv\_d^2 \lambda\_0}{2\pi \epsilon\_0 c^2 r} \,\tag{62}$$

since *I* = *λ*−*vd* = *λ*0*vd*. In the electron frame, the speed of *q* is zero so the force is purely electrostatic:

$$F' = qE' = \frac{q\lambda'}{2\pi\epsilon\_0 r} = \frac{\gamma q v\_{d'}^2 \lambda\_0}{2\pi\epsilon\_0 c^2 r}.\tag{63}$$

The electron frame is the rest frame for *q*; hence (47) becomes

$$F = F'/\gamma.\tag{64}$$

which is obviously satisfied. Hence, what appears as a purely magnetostatic force in the ion frame appears as a purely electrostatic force in the electron frame.

In the general case, where a point-like object of charge *q* is moving at any speed *v* (say, in the ion frame), the ions and electrons are contracted by different factors, but always in such a way that the resulting net electrostatic force is balanced by the net magnetostatic force between the charge *q* and both on and electron currents. This case is discussed in detail by Gabuzda (1987).

#### **5.4 Two parallel wires**

As a final case, consider two parallel wires, each carrying a current *I*. When considering the effect of wire 1 on wire 2, we must consider the electrons and ions in wire 2 separately, as they have no common rest frame (Redži´c, 2010). We cannot reify one segment of length *l* in one frame and transform it as a whole, even though coincidentally the same formulae can be obtained (van Kampen, 2008; 2010).

One convenient way of looking at the problem is by considering a segment of wire 2 of length *l*, as measured in the ion frame. This consists of a segment of ions of length *l*/*γ* as seen in the electron frame, and a segment of electrons of length *γl* as seen in the electron frame. The transformation leaves the total charge unaltered: it is *λ*0*l*.

In the ion frame, we can find the total force on the ion segment by dividing it up into point-like segments of charge density *λ*0, and integrating over the entire length *l*. An identical procedure holds for the electrons. In the electron frame, we integrate ion segments of charge density *γλ*<sup>0</sup> over a length *l*/*γ*, and electron segments of charge density *λ*0/*γ* over a length *γl*. The net result is that all expressions found in the previous paragraph hold, if we replace *q* with *λ*0*l*:

$$F\_{\mathfrak{e}+} = F\_{\mathfrak{m}+} = F\_{\mathfrak{e}-} = F\_{\mathfrak{m}-}' = \mathbf{0} \tag{65}$$

$$F\_{\varepsilon+}^{\prime} = F\_{\varepsilon-}^{\prime} = F\_{m+}^{\prime} = \frac{\gamma \lambda\_0^2 v\_d^2 l}{2\pi \epsilon\_0 c^2 r} \tag{66}$$

$$F\_{m-} = \frac{\lambda\_0^2 v\_d^2 l}{2\pi \epsilon\_0 c^2 r} \tag{67}$$

#### **6. Conclusion**

16 Will-be-set-by-IN-TECH

must make sure that the effect on charges near the wire is the same in both frames; otherwise,

First, consider a point-like object of charge *q* stationary in the ion frame. There is no electrostatic force on the object, since there is no net charge; nor is there a magnetostatic force,

*<sup>e</sup>* <sup>=</sup> *qE*� <sup>=</sup> *<sup>q</sup>λ*0*γv*<sup>2</sup>

*<sup>m</sup>* <sup>=</sup> *qvdB*� <sup>=</sup> *qvdμ*<sup>0</sup> *<sup>I</sup>*

forces cancel, because a current of positive ions attracts a positively charged object while the

As a second case, a point-like object of charge *q*, moving parallel to a current carrying wire at speed *vd* in the ion frame, experiences a purely magnetostatic force due to the electrons in the

since *I* = *λ*−*vd* = *λ*0*vd*. In the electron frame, the speed of *q* is zero so the force is purely

*F* = *F*�

which is obviously satisfied. Hence, what appears as a purely magnetostatic force in the ion

In the general case, where a point-like object of charge *q* is moving at any speed *v* (say, in the ion frame), the ions and electrons are contracted by different factors, but always in such a way that the resulting net electrostatic force is balanced by the net magnetostatic force between the charge *q* and both on and electron currents. This case is discussed in detail by Gabuzda (1987).

As a final case, consider two parallel wires, each carrying a current *I*. When considering the effect of wire 1 on wire 2, we must consider the electrons and ions in wire 2 separately, as they have no common rest frame (Redži´c, 2010). We cannot reify one segment of length *l* in one frame and transform it as a whole, even though coincidentally the same formulae can be

<sup>2</sup>*π<sup>r</sup>* <sup>=</sup> *qv*<sup>2</sup>

<sup>2</sup>*π�*0*<sup>r</sup>* <sup>=</sup> *<sup>γ</sup>qv*<sup>2</sup>

*<sup>d</sup>λ*<sup>0</sup> 2*π�*0*c*2*r*

*<sup>d</sup>λ*<sup>0</sup> 2*π�*0*c*2*r*

*<sup>F</sup>* <sup>=</sup> *qvdB* <sup>=</sup> *qvdμ*<sup>0</sup> *<sup>I</sup>*

*<sup>F</sup>*� <sup>=</sup> *qE*� <sup>=</sup> *<sup>q</sup>λ*�

*d*

<sup>2</sup>*π�*0*c*2*r*� . (60)

<sup>2</sup>*πr*� . (61)

<sup>+</sup>*vd* = *γλ*0*vd* and *<sup>μ</sup>*<sup>0</sup> = 1/*�*0*c*2. The

, (62)

. (63)

/*γ*, (64)

*<sup>m</sup>*, as in the electron frame the object is moving with speed

*<sup>e</sup>* is given

because the speed of the object is zero. In the electron frame, the electrostatic force *F*�

*F*�

*F*�

the principle of relativity would be violated.

There is also a magnetostatic force, *F*�

positive charge density repels it.

wire, given by:

electrostatic:

**5.4 Two parallel wires**

obtained (van Kampen, 2008; 2010).

*vd* parallel to a current of positive ions; hence

The two forces are readily shown to be equal, as *I* = *λ*�

The electron frame is the rest frame for *q*; hence (47) becomes

frame appears as a purely electrostatic force in the electron frame.

by

In this chapter, we have outlined how electrostatics, magnetostatics and special relativity give consistent results for a few cases involving infinitely long current carrying wires. We have used an online integrator to obtain expression for the electrostatic field due to a hollow uniformly charged cylinder, and derived expressions for a solid uniformly charged cylinder and current-carrying wires from it. We have also derived expressions for self-pinching in a current-carrying wire, by a factor *γ*, and the creation of a surface charge density.

#### **7. Acknowledgement**

The author gratefully acknowledges fruitful discussions with Enda McGlynn and Mossy Kelly.

#### **8. References**


**1. Introduction**

Classical electromagnetism is a well-established discipline. However, there remains some confusions and misunderstandings with respect to its basic structures and interpretations. For example, there is a long-lasting controversy on the choice of unit systems. There are also the intricate disputes over the so-called EH or EB formulations. In some textbooks, the authors respect the fields *E* and *B* as fundamental quantities and understate *D* and *H* as auxiliary

**Reformulation of Electromagnetism with**

These confusions mainly come from the conventional formalism of electromagnetism and also from the use of the old unit systems, in which distinction between *E* and *D*, or *B* and *H* is blurred, especially in vacuum. The standard scalar-vector formalism, mainly due to Heaviside, greatly simplifies the electromagnetic (EM) theory compared with the original formalism developed by Maxwell. There, the field quantities are classified according to the number of components: vectors with three components and scalars with single component. But this classification is rather superficial. From a modern mathematical point of view, the field quantities must be classified according to the tensorial order. The field quantities *D* and *B* are the 2nd-order tensors (or 2 forms), while *E* and *H* are the 1st-order tensors (1 forms).

The constitutive relations are usually considered as simple proportional relations between *E* and *D*, and between *B* and *H*. But in terms of differential forms, they associate the conversion of tensorial order, which is known as the Hodge dual operation. In spite of the simple appearance, the constitutive relations, even for the case of vacuum, are the non-trivial part of the EM theory. By introducing relativistic field variables and the vacuum impedance, the

The EM theory has the symmetry with respect to the space inversion, therefore, each field quantity has a definite parity, even or odd. In the conventional scalar-vector notation, the parity is assigned rather by hand not from the first principle: the odd vectors *E* and *D* are named the polar vectors and the even vectors *B* and *H* are named the axial vectors. With respect to differential forms, the parity is determined by the tensorial order and the pseudoness (twisted or untwisted). The pseudoness is flipped under the Hodge dual operation. The way of parity assignment in the framework of differential forms is quite

quantities. Sometimes the roles of *D* and *H* in a vacuum are totally neglected.

(The anti-symmetric tensors of order *n* are called *n*-forms.)

constitutive relation can be unified into a single equation.

natural in geometrical point of view.

*Japan*

Masao Kitano

*Kyoto University,*

**Differential Forms**

*Department of Electronic Science and Engineering,*

**0**

**2**

