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15.1: Introduction

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  • One of Newton's great achievements was to show that all of the phenomena of classical mechanics can be deduced as consequences of three basic, fundamental laws, namely Newton's laws of motion. It was likewise one of Maxwell's great achievements to show that all of the phenomena of classical electricity and magnetism – all of the phenomena discovered by Oersted, Ampère, Henry, Faraday and others whose names are commemorated in several electrical units – can be deduced as consequences of four basic, fundamental equations. We describe these four equations in this chapter, and, in passing, we also mention Poisson's and Laplace's equations. We also show how Maxwell's equations predict the existence of electromagnetic waves that travel at a speed of \(3 \times 10^8\, \text{m} \,\text{s}^{-1}\). This is the speed at which light is measured to move, and one of the most important bases of our belief that light is an electromagnetic wave.

    Before embarking upon this, we may need a reminder of two mathematical theorems, as well as a reminder of the differential equation that describes wave motion.

    The two mathematical theorems that we need to remind ourselves of are:

    • The surface integral of a vector field over a closed surface is equal to the volume integral of its divergence.
    • The line integral of a vector field around a closed plane curve is equal to the surface integral of its curl.

    A function \(f(x-v t)\) represents a function that is moving with speed \(v\) in the positive \(x\)-direction, and a function \(g(x-v t)\) represents a function that is moving with speed \(v\) in the negative \(x\)-direction. It is easy to verify by substitution that \(y=Af+Bg\) is a solution of the differential equation

    \[ \dfrac{d^2y}{dt^2} = v^2 \dfrac{d^2y}{dx^2} \tag{15.1.1} \label{15.1.1}\]

    Indeed it is the most general solution, since \(f\) and \(g\) are quite general functions, and the function \(y\) already contains the only two arbitrary integration constants to be expected from a second order differential equation. Equation \ref{15.1.1} is, then, the differential equation for a wave in one dimension. For a function \(\Psi(x,y,z)\) in three dimensions, the corresponding wave equation is

    \[ \ddot \Psi = v^2 \nabla^2 \Psi \tag{15.1.2}\]

    It is easy to remember which side of the equation \(v^2\) is on from dimensional considerations.

    One last small point before proceeding – I may be running out of symbols! I may need to refer to surface charge density, a scalar quantity for which the usual symbol is \(\sigma\). I shall also need to refer to magnetic vector potential, for which the usual symbol is \(\bf{A}\). And I shall need to refer to area, for which either of the symbols \(A\) or \(\sigma\) are commonly used – or, if the vector nature of area is to be emphasized, \(\bf{A}\) or \(\boldsymbol{\sigma}\). What I shall try to do, then, to avoid this difficulty, is to use \(\bf{A}\) for magnetic vector potential, and \(\sigma\) for area, and I shall try to avoid using surface charge density in any equation. However, the reader is warned to be on the lookout and to be sure what each symbol means in a particular context.