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

The net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere is (kQ²)/(2R²), where k is the electrostatic constant, Q is the total charge on the sphere, and R is the radius of the sphere.

Step-by-step explanation:

The net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere can be found by considering the electric field at the surface of the sphere. Since the charge distribution is spherically symmetric, the electric field at the surface of the sphere will only have a radial component. Using Gauss's law, we can determine that the electric field at the surface is given by E = kQ/R², where k is the electrostatic constant, Q is the total charge on the sphere, and R is the radius of the sphere.

To find the force, we can multiply the electric field by the charge on the northern hemisphere. The charge on the northern hemisphere can be calculated as half the total charge on the sphere, Q/2. Therefore, the net force is given by F = (kQ²)/(2R²).

Answer 2

Step-by-step 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 = (1)/(4\pi epsilon) (Qr)/(R^(3) )`

dq = ρdV =
`(3Q)/(4\pi R^(3) )`
`r^(2)`dr.sinθdθdΦ

integrate it over half ball,

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

=
`(3Q^(2) )/(32\pi epsilonR^(2) ) \int\limits^(\pi )/(2) _b {} \,` sinθcosθdθ

=
`(3Q^(2) )/(64\pi epsilon R^(2) )`

`F = (3Q^(2) )/(64\pi epsilon R^(2) ) z`