Question

find the electric field at a distance Z above the centre of a square loop ( side a) carrying uniform line change density.​

Answer 1

Answer:

So the problem is asking me to find the Electric field a height z above the center of a square sheet of side a

I approach the problem a different way than the book, I derive the electric field due to a line of charge of side a a height z above the center of a square loop, and I verified it to be 14πϵ0 λaz(z2+a24)(z2+a22)12 z^

Now the way I do it is that I let that line have a thickness da where da is a width element not an area element (as the side of the square is a), so now the linear charge density λ is equal to the surface charge density multiplied by that small thickness da , that is

λ=σda

So the Electric field dE due to a line of small thickness da is

dE = 14πϵ0 σdaza(z2+a24)(z2+a22)12 z^

I integrate this field from 0 to a then,

E = σz4πϵ0 ∫a0 ada(z2+a24)(z2+a22)12 z^

This integral yields 4z tan−1(1+a22z2−−−−−−√ |a0

= 4z [tan−1(1+a22z2−−−−−−√−π4]

That is the value of the integral, now multiply it by σz4πϵ0

Then E=σπϵ0 [tan−1(1+a22z2−−−−−−√−π4]

I'm missing it by a factor of 2, the answer should be 2σπϵ0 [tan−1(1+a22z2−−−−−−√)−π4]

Explanation:

mark me brainlest