Question

Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radius R and the total charge Q.

Answer 1

Answer:

Explanation:

The final net force will be in the Z- direction. Let's find out the z component of the force on the differential volume of charge is:

df = dqEcosθz

$E = \frac{1}{4\pi epsilon} \frac{Qr}{R^{3} }$

dq = ρdV =  $\frac{3Q}{4\pi R^{3} }$$r^{2}$dr.sinθdθdΦ

integrate it over half ball,

$F_{z} = \int\limits^._V {df_{x}dV} =\frac{1}{4\pi epsilon } \frac{Q}{R^{3} } \frac{3Q}{4\pi R^{3} }\int\limits^R_0 {\int\limits^\frac{\pi }{2} _{0} {\int\limits^\frac{\pi }{2} _0 {r^{3} } \, dr } \, } \,$.sinθcosθdθdΦ.( these are part of the integral, i was unable to write it in equation format).

    = $\frac{3Q^{2} }{32\pi epsilonR^{2} } \int\limits^\frac{\pi }{2} _b {} \,$  sinθcosθdθ

   

   = $\frac{3Q^{2} }{64\pi epsilon R^{2} }$

$F = \frac{3Q^{2} }{64\pi epsilon R^{2} } z$