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1.6B: Spherical Charge Distributions

I shall not here give calculus derivations of the expressions for electric fields resulting from spherical charge distributions, since they are identical with the derivations for the gravitational fields of spherical mass distributions in the Classical Mechanics “book” of these physics notes, provided that you replace mass by charge and \(G\) by -1 /(4π\(\epsilon_0\)) . See Chapter 5, subsections 5.4.8 and 5.4.9 of Celestial Mechanics. Also, we shall see later that they can be derived more easily from Gauss’s law than by calculus. I shall, however, give the results here.

At a distance \(r\) from the centre of a hollow spherical shell of radius a bearing a charge \(Q\), the electric field is zero at any point inside the sphere (i.e. for \(r < a\)). For a point outside the sphere (i.e. \(r > a\)) the field intensity is

\[E=\frac{Q}{4\pi\epsilon_0 r^2}.\tag{1.6.4}\]

This is the same as if all the charge were concentrated at a point at the centre of the sphere.

If you have a spherically-symmetric distribution of charge \(Q\) contained within a spherical volume of radius a, this can be considered as a collection of nested hollow spheres. It follows that at a point outside a spherically-symmetric distribution of charge, the field at a distance \(r\) from the centre is again

\[E=\frac{Q}{4\pi\epsilon_0 r^2}.\tag{1.6.5}\]

That is, it is the same as if all the charge were concentrated at the centre. However, at a point inside the sphere, the charge beyond the distance \(r\) from the centre contributes zero to the electric field; the electric field at a distance \(r\) from the centre is therefore just

\[E=\frac{Q_r}{4\pi\epsilon_0 r^2}.\tag{1.6.6}\]

Here \(Q_r\) is the charge within a radius \(r\). If the charge is uniformly distributed throughout the sphere, this is related to the total charge by \(Q_r=\left ( \frac{r}{a}\right )^3 Q\), where \(Q\) is the total charge. Therefore, for a uniform spherical charge distribution the field inside the sphere is

\[E=\frac{Qr}{4\pi\epsilon_0 a^3}.\tag{1.6.7}\]

That is to say, it increases linearly from centre to the surface, where it reaches a value of \(\frac{Q}{4\pi\epsilon_0 a^2}\), whereafter it decreases according to equation 1.6.5.

It is not difficult to imagine some electric charge distributed (uniformly or otherwise) throughout a finite spherical volume, but, because like charges repel each other, it may not be easy to realize this idealized situation in practice. In particular, if a metal sphere is charged, since charge can flow freely through a metal, the self-repulsion of charges will result in all the charge residing on the surface of the sphere, which then behaves as a hollow spherical charge distribution with zero electric field within.