**3. Maxwell's equations**

The discoveries made by scientists over the past millennia have contributed towards the profound understanding of electromagnetics that we have today. Among these scientific discoveries, it is the experimental observations reported independently by Ampere, Faraday and Gauss which inspired James Clerk Maxwell to establish the unified theory of electricity and magnetism. In 1873, Maxwell published his formulations in his textbook *A Treatise on Electricity and Magnetism*. The complexity of the formulations in the textbook was later reduced by Oliver Heaviside in 1881 to four sets of differential equations.

 More popularly referred to as Maxwell's equations today, these four sets of notable mathematical equations which outline the fundamental principles of electromagnetism are tabulated in **Table 1**. Eq. (2.1) in the table describes the observation reported by the English physicist, Michael Faraday. According to Faraday, when the magnetic field intensity (**H**) or magnetic flux density (*μ***H**) varies with time (*t*), a force will be induced, where **E** is the electric field intensity and the line integral is performed over a line contour *C* bounding an arbitrary surface. Established by the French physicist Andre-Marie Ampere, Ampere's circuital law in Eq. (2.2) states that the circulation of **H** around a closed loop will result


**Table 1.**  *Maxwell's equations.* 

in current traversing through the surface bounded by the loop. Here, **J** is the convection current density, while is the displacement current density. Eqs. (2.3) and (2.4) are formulated based on the observations made by the German physicist, Carl Friedrich Gauss. According to Gauss, the total electric flux density (ε**E**) flowing out from an enclosed surface *S* is equivalent to the total charge (ρ) encapsulated within *S*, i.e. (2.3), whereas the net outward **H** from *S* is invariably zero, i.e. Eq. (2.4).
