Physics Friday 14

Find the electric field a distance z above the center of a flat circular disk of radius R, which carries a uniform surface charge σ. What does this approach in the limit  and in the limit .

Let us set up our coordinates. We have a disk of radius R in the xy plane centered at the origin, with uniform surface charge σ, and are observing the field at a point on the z axis, z>0. Letting  be the displacement vector from a given charge element to the point where we observe the field, then the field is given by:


We note that for a total charge on our disk of Q, we have . Thus , and so .

Next, we note that the system has rotational symmetry about the z axis, and thus we find that the eletcric field must be directed along the z axis; we can ignore the x and y components in the integral, as they cancel. Working in cylindrical coordinates (r, θ, z), we have for the xy plane . For a charge element at (r, θ, 0), the displacement vector  has a z-component of z, and a magnitude .
So we obtain the integral:






Now, in the limit , we have:
,
and when , , so we obtain:
,
or, using , we get
, which is the field a distance z from a point charge Q, as one might expect.

In the limit , we use  to obtain

which is the field for an infinite plane of uniform charge area density σ.