🚀 go-pugleaf

On Thu, 1 Jun 1995, Robert J. Devantery wrote:

> I have been having a discussion with another physics teacher who believes that
> there is no voltage inside the sphere of a Van de Graaff generator.  I
> know that the electric field is zero but believe the voltage is constant
> and is the same as the surface value.

Bob,

So *how* does one know the electric field is zero inside? First consider
a *solid* conducting sphere which is given an excess charge, and achieves
static equilibrium. If there were any potential difference between any
two points within the sphere, any excess charges there would be in motion,
which contradicts our assumption of static equilibrium of the excess charge.

When excess charge is placed on or within such a sphere, each charge
carrier moves under the combined field of all the others, which is to the
surface. During this time the fields within the sphere may be quite
complicated, depending where you put the excess charge. But finally an
equilibrium is established in which the fields and the charges settle
down to a situation in which the fields cannot move the charges anywhere.
That situation is with uniform charge density over the surface of the
sphere, and the electric fields normal to the surface of the sphere. And
no field inside.

Once such a static situation is established, one can imagine removing the
inner volume of the conducting sphere, leaving only a spherical shell on
which the excess charge resides. Nothing is changed, and the field inside
is still zero.

The fact that the field must be normal to the conductor surface *when a
static charge distribution is achieved* is easily demonstrated. If it were
*not* normal to the surface, it would have a component tangent to the
surface and any charge at the surface would experience a force moving it
across the surface, which contradicts our statement that this was a static
distribution.

Potential difference between points A and B is defined as the work/charge
done against the electric field in moving a charge from A to B. Since
there's no field inside the sphere, moving a charge between any two points
within the sphere requires no work, hence the potential difference
between the points must be zero. If one of the points is on the surface
of the sphere, the other point within the sphere must have the same
potential as the surface of the sphere. Therefore the potential has a
uniform, constant, nonzero value within and on the sphere.

However, in the case of the *running* Van de Graff, the situation is not
strictly static. Charge is leaking to the air around the sphere and being
replaced by charge brought up by the belt. Therefore there must be some
potential difference between the belt and the surface of the sphere, or
the charge could never get from the belt to the surface. So the potential
is not quite uniform within this sphere.

        -- Donald