Field from a spherical shell, right and wrong ** The electric field outside and

Question

Field from a spherical shell, right and wrong ** The electric field outside and an infinitesimal distance away from a uniformly charged spherical shell, with radius R and surface charge density sigma , is given by Eq. (1.42) as sigma/elementof_0. Derive this in the following way. (a) Slice the shell into rings (symmetrically located with respect to the point in question), and then integrate the field contributions from all the rings. You should obtain the incorrect result of sigma/2 elementof_0. (b) Why isn't the result correct? Explain how to modify it to obtain the correct result of sigma/elementof_0. Does the above integration provide a good description of what's going on for points on the shell that are very close to the point in question?

Explanation / Answer

a. consider a uniformly charged shell of radius R, and surface charge density sigma

consider s point at a distance R from the center of ths shell

so splitting the shell into rings perpendicular to the axis of ths shell ( the line joining th epoint of interest with the center of the shell)

radius of an arbitrary shell, r = Rsin(theta)

distance from the point , d = R(1 - cos(theta))

surface area, dA = 2*pi*r*R*d(theta)

so charge on this ring, dq = sigma*dA

hence, electric field due to this ring at point is dE = kd*dq/(d^2 + r^2)^3/2

dE = k*R(1 - cos(theta))*sigma*2*pi*r*R*d(theta)/((R(1 - cos(theta)))^2 +( Rsin(theta))^2)^3/2

dE = 2*pi*k*sigma(1 - cos(theta))*sin(theta)*d(theta)/(2 - 2cos(theta))^3/2

integrating from theta = -90 to +90 we get

E = sigma/2*epsilon

b. The result is incorrect as the value obtained by gauss' law is something else. the result might well be incorrect becasuse of the assumption of trying to find the field at point R

This approximation getsd us the exact integral at the shell surface and not inside it ( or outside it)

to get the right result, we should find electric field just inside the shell, and see if it comes out to be 0, if it does not, then we have to subtract this from the prevous result to get the right answer

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