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*.

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Tags: Electrostatics, Friday Physics, Science

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