Week 2: Continuous Charge and Gauss’s Law 89
In a typical conductor – for example a metal such as silver or copper – there is on average roughly
one free electronper atomin the material. That is in the ballpark of 10^24 free electrons per mole of
metal, which in turn is somewhere between 10^4 and 10^5 Coulombs of free charge! As we discussed in
class, two charges of one Coulomb each separated by one meter exert a force of 9× 109 Newtons on
each other, more than enough torip apart any material known to mankind. Consequently we have
no hope of either removing all of the free electrons from a piece of metal and separating them by
any appreciable macroscopic distance, or adding enough electronsso that every atom has an extra
one. The material would come apart long before we succeeded.
This means that we can consider the free charge in a conductor to beinexhaustible. As far as
we’re concerned, we can always add charge to a conductor, or take it away, or rearrange it as we
please with fields and forces, and never run a risk of “saturating” the conductor’s ability to supply
still more free charge, at least not as long as the conductor remains intact.
Now let’s think a moment about the “free” bit. If we exert a force onthe charges in a conductor
(with, say, an electric field), they are free to move and hence will accelerate in the direction of the
force. They will continue to move, speeding up, until they encounter an insulated boundary of the
material, where they must stop. There they build up until they create a field of theirownthat
cancelsthe applied external field, at least inside the conductor. Eventuallythe conductor can reach
a state ofstatic equilibriumwhere all the forces on all of the charges, including a “surface force”
that holds the mobile charges inside the conductor at the surface,cancel.
When the conductor is in static equilibrium, we can then conclude the following:- The electric field inside a conductor in static equilibrium vanishes.If the field were
not zero, it would exert a force on the free charges inside the conductor. Since they’re free,
they’d move. If they move, they’re not in equilibrium. So the field mustbe zero. - The electric field parallel to the surface of the conductor instatic equilibrium
vanishes. The same argument. If there were a field component parallel to thesurface, it
would exert a force on charges on the surface, they can move (parallel to the surface) and
hence would move, contradicting the assumption of equilibrium. Notethat this doesnot
restrict the fieldperpendicularto the surface of the conductor! - The electric field just outside of the surface of a conductor in electrostatic equi-
librium is perpendicular to the surface.Furthermore, from Gauss’s Law we can see that
it must be true that: - E⊥= 4πkeσwhereσis the charge per unit area on the surface of the conductor.
To prove this, consider a Gaussian pillbox (drawn in figure 22 above) that barely encloses the
surface. Inside, the field is zero so the flux through the inside pillboxlid vanishes. The flux
through the sides is zero because there is no field parallel to the sides. The flux through the
outerpillbox surface only must therefore equal the charge inside:
E⊥A= 4πkeQS= 4πkeσA (121)and the result is proven.- There can be no surplus charge inside a conductor in electrostatic equilibrium.
This follows from Gauss’s Law in reverse. We noted above that the field must vanish inside
a conductor in equilibrium. This means that the flux through any closed surface drawn com-
pletely inside the conductor must vanish. This means in turn that thenet charge inside that
surface must vanish for all possible surfaces, which suffices to prove that there can be no net
charge inside the conductor.
As a corollary, any unbalanced charge on a conductor in equilibriummust be found on the
surfaceand must, of course, be related toE~⊥at the surface.