A sphere of radius R carries a polarization P(r)=krsin(theta) {vector sign above

Question

A sphere of radius R carries a polarization P(r)=krsin(theta)   {vector sign above p and r} where k is a constant and r(vector sign above it) is the vector from the center.  (a) Calculate the bound charges ?b and ?b .(b) Calculate the total bound charge on the sphere.
A sphere of radius R carries a polarization P(r)=krsin(theta)   {vector sign above p and r} where k is a constant and r(vector sign above it) is the vector from the center.  (a) Calculate the bound charges ?b and ?b .(b) Calculate the total bound charge on the sphere.

Explanation / Answer

a)

bound charge

= - div.P

=- k sin theta + k cos theta

= k (cos theta - sin theta)

b)

total bound charge

= int (-div P)* 2pi r^2 dr sin theta d theta

= [k *2 pi r^3]/3 * [cos theta - sin theta] sin theta d theta

theta varies form 0 to pi

carry out this integration

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