Finite Sheets Electric Field question

I was trying to solve this problem but was unable to...I hope someone here could assist me with it.

Consider two thin disks, of negligible thickness, of radius

oriented perpendicular to the x axis such that the x axis runs through the center of each disk. The disk centered at

has positive charge density

, and the disk centered at

has negative charge density $-\eta$ , where the charge density is charge per unit area.

What is the magnitude

of the electric field at the point on the x axis with x coordinate

?
Express your answer in terms of

, and the permittivity of free space

Reply 1

Morbo

Consider a disc as you have described in the y-z plane, perpendicular to the x-axis with its centre at the origin, charge density n, radius R.

Now find the potential at a point x along the x-axis, which is the sum of the potential from all the charge area elements on the disc, which are given by n*(r dr dt), where r is the distance along a radius on the disc, and t is the angle from the y-axis.

dV_x = dq/4*pi*e0*|a| = (n*r dr dt)/[4*pi*e0*sqrt(x² + r²)]

∫dV_x = V_x = n/(4*pi*e0) * ∫₀^2pidt * ∫₀^R r dr/sqrt(x² + r²)

V_x= n/(4*e0) * sqrt(x² + R²)

So, the electric field along the x-axis is given by E_x = -dV_x/dx:

|E_x| = [n*x/(2*e0)] / sqrt(x² + R²)

So this gives you the electric field at an axial distance x from the disc. Hope this helps you.