**Fundamentals**

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*Ireland*

Paul van Kampen

*School of Physical Sciences, Dublin City University*

**Current-Carrying Wires and Special Relativity**

*Centre for the Advancement of Science and Mathematics Teaching and Learning &*

This chapter introduces the main concepts of electrostatics and magnetostatics: charge and current, Coulomb's Law and the Biot-Savart Law, and electric and magnetic fields. Using linear charge distributions and currents makes it possible to do this without recourse to vector calculus. Special relativity is invoked to demonstrate that electricity and magnetism are, in a sense, two different ways of looking at the same phenomenon: in principle, from a knowledge of either electricity or magnetism and special relativity, the third theory could be derived. The three theories are shown to be mutually consistent in the case of linear currents and charge

This chapter brings together the results from a dozen or so treatments of the topic in an internally consistent manner. Certain points are emphasized that tend to be given less prominence in standard texts and articles. Where integration is used as a tool to deal with extended charge distributions, non-obvious antiderivatives are obtained from an online integrator; this is rarely encountered in textbooks, and gives the approach a more contemporary feel (admittedly, at the expense of elegance). This enables straightforward derivation of expressions for the electric and magnetic fields of radially symmetric charge and current distributions without using Gauss' or Ampère's Laws. It also allows calculation of the extent of "self-pinching" in a current-carrying wire; this appears to be a new result.

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

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

**1. Introduction**

distributions.

**2. Electrostatics**

force big enough to overcome this pull (Arons, 1996).

**2.1 Charge**
