Please show all steps clearly and easy to follow. E3A.1 Consider a spherical she

Question

Please show all steps clearly and easy to follow.

E3A.1 Consider a spherical shell of radius R with charge Q uniformly distributed on its surface. Our goal is to prove the shell theorerm by calculating the potential (r) at a point P that is a distance r from the shell's center. Do this by dividing the shell's surface into thin rings, where the ith ring is the set of points on the shell's surface between the "latitudes 2 and + 2 , where the shell's north pole" is the point on the shell closest to point P and we measure the "latitude" angle from that pole (as is ntional in physics) rather than from the equator. First show that the ith ring has charge qi = 10sm ,A0. Then find the potential , contributed at P by this ring, sum over all rings, convert the sum to an integral, and evaluate the integral for both r > R and r

Explanation / Answer

given
spherical shell, radius = R
charge = Q, uniformly distributed on the surface

now, consider ith ring
its present between latitudes thetai - 0.5theta and thetai + 0.5*theta

hence charge on it is given by
qi = (Q/4*pi*R^2) * 2*pi*r*x
x is thixkness of the strip , x = theta*R
r is radius of this ring, r = Rsin(theta)
hence
qi = Q*theta*sin(theta)/2

for theta = d(theta)
qi = Q*sin(theta)d(theta)/2

potential of this ring at point r is
phii = k*qi/sqroot(R^2*sin^2(theta) + (r - Rcos(theta))^2)
phii = k*qi/sqroot(R^2 + r^2 - 2rRcos(theta))

hence total potential
integrate phii from theta = 0 to theta = pi
hence
phi = kQ(integrate(sin(theta)*d(theta)/sqroot(R^2 + r^2 - 2rRcos(theta))))/2

integrating we get
phi = kQ(cos(0)/sqroot(R^2 + r^2 - 2rRcos(0)) - cos(180)/sqroot(R^2 + r^2 - 2rRcos(180)))/2
for r > R
phi = kQ(1/(r - R) + 1/(R + r)) = kQ*r/(r^2 - R^2)

for R > r
phi = kQ(1/(R - r) + 1/(R + r)) = kQ*R/(R^2 - r^2)

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