12.7.1 Charged Metal Spheres

Alright, it’s time to graduate from point charges. To be honest, engineers seldom work with individual point charges. More often we work with lines or areas of charges, which are commonly found in charged metallic conductors.

Let’s consider a neutral metallic sphere, which we know contains many moles of electrons. Due to the mutual repulsion among them, these electrons would rather be as far apart from one another as possible. But alas, they cannot escape into the surrounding because they are still bound to the positively charged ions that make up lattice. So the electrons do the best they can do, which is to spread themselves uniformly throughout the volume of the sphere. As a result, the sphere is neutral in charge throughout and there is zero electric field everywhere in the sphere.

Now let’s bring some electrons from elsewhere and drop them into the sphere. The sphere as a whole is now negatively charged. But the electrons will work out a new way to distribute themselves so that the electric field throughout its volume remains zero. Unlike their counterparts in an insulator, the electrons in a conductor are not bound to any specific ion and are free to roam within the metal. If the net field is not zero anywhere inside the sphere, the electrons at those positions will experience a non-zero net force, and must therefore be still sloshing around. The fact that the electrons have “settled down” means that an equilibrium state has been achieved and the net field is zero everywhere in the sphere.

What happens in the charged sphere is that the surplus electrons are pushed outward towards the edges: the neutrality within the sphere is restored, but the surface of the sphere now attains a negative charge. This distribution works because a spherical shell of charges produces zero field everywhere inside the shell (google “shell theorem” if you’re interested).

Similarly, we can remove some of the electrons and make the sphere positively charged as a whole. What the electrons do now is to close ranks and remove any “gaps” in the sphere so that it remains neutral in charge everywhere in the sphere. The surface of the sphere, abandoned by the electrons, now attains a positive charge. Again, this solution is legit because a spherical shell of charges produces zero field everywhere inside the shell.

Electrons in irregularly shaped conductors will have to come up with less obvious distributions to achieve $\displaystyle E=0$ in the conductor. Nevertheless, whatever the shape of the conductor, the following outcomes are true for any conductor at equilibrium:

1) The net electric field in a conductor is zero.
Else the electrons in the conductor would not be at rest.

2) Any unbalanced electric charge must reside on its surface.
Else the net field in the conductor cannot be zero.

3) The electric potential is uniform throughout the conductor.
Because $\displaystyle E=0$ in the conductor implies zero potential gradient in the conductor.

4) The surface of the conductor is an equipotential surface.

5) The external electric field at the surface of the conductor is perpendicular to the surface.
Because $\displaystyle E=-\frac{{dV}}{{dr}}$ , field lines must cut equipotential lines perpendicularly.

In fact, because the electrons are abundant and mobile in a conductor, even when an external electric field is applied, the electrons in a conductor will not rest until they find a way to distribute themselves so that the net field everywhere in the conductor is zero. It is amazing the electrons can always “compute” the solution in a split second.

For example, let’s have a rectangular slab of metal. When a rightward external electric field is applied across the slab, the electrons in the conductor will be pulled to the left face of the slab. This results in negative charges on the left face, and positive charges on the right face. These induced charges will now produce a leftward internal electric field of its own. At equilibrium, the electrons in the slab will distribute themselves such that the external field is balanced by the internal field. Only then, will the electrons in the slab stop moving.

Example

Consider a charged metal sphere.

a) Draw the field lines and equipotential lines to illustrate the electric field of a charged metal sphere.

b) Sketch graphs to show the variation of E and V along a line passing through the centre of the sphere.

Solution

Note that

There should be no field lines within the metal sphere.
Outside the sphere, the charged sphere is equivalent to a point charge at the centre of the sphere.
Inside the sphere, $\displaystyle E=0$ and $\displaystyle V=\text{constant}$ .