Electric field due to a flat circular disk

[grandpa2390]

Homework Statement

Find the electric field at a distance z above the center of a flat circular disk of radius R

Homework Equations

The Attempt at a Solution

My attempt to solve this was take the line integral from the center of the circle to the edge. Then, knowing the circle is symmetrical, multiply it by 2*π*r to get the entire circle. 2πr of my line charges make a circle.
that approach worked when finding the electric field due to a ring. In which case I found the field due to a point charge, and multiplied it by the circumference because 2*π*r of the points made the ring. so 2*π*r of the lines should make the area...

but I am not getting the correct answer. On Chegg, They found the an equation for a ring (with radius dr) and integrated. I definitely see how that approach should work, it is the same idea as mine but in reverse (sort of).

My questions is: is my approach wrong? or am I making a mistake in my work?

 

[kuruman]

I am not sure what you mean by line integral. You need to do a surface integral. Are you saying that you calculated the contribution to the z-component of the field from a pie-shaped piece of angle Δφ? If that's what you did, and did correctly, it should work because all you have to do is set Δφ = 2π.

 

[grandpa2390]

I am not sure what you mean by line integral. You need to do a surface integral. Are you saying that you calculated the contribution to the z-component of the field from a pie-shaped piece of angle Δφ? If that's what you did, and did correctly, it should work because all you have to do is set Δφ = 2π.

when I did the field due to a ring, I found the E field due to a point and then multiplied it by the circumference.

I tried to do the same thing with the disk, except instead of a the E field from a point at the edge, I found the E field from all the points between the center and the edge in a straight line. Then I multiplied that line by 2πr

but what I am thinking now is that this approach would probably overcount the area of the circle. as there are less points closer to the center of the circle than the outer circle.

 

[haruspex]

when I did the field due to a ring, I found the E field due to a point and then multiplied it by the circumference.

That does not work, unless you mean that you found the z component of the E field due to a point.

this approach would probably overcount the area of the circle. as there are less points closer to the center of the circle than the outer circle.

Quite so. And it is not just a question of overcounting the area as a whole, but of biasing towards the contribution from the central parts.

Having found the field due to a ring, integrate in the radial direction.