Question

A conducting sphere of radius R carries positive charge q. Calculate the amount of work that would be required to move a small positive test charge q0 slowly from r=5R to r=3R. Assume that the presence of q0 has no effect on how the charge q is distributed over the sphere.
Express your answer in terms of the electric constant ϵ0 and some or all of the variables q, q0, and R.

Answer 1

Answer:

Explanation:

Given

Radius of Conducting Sphere is R with Positive charge q

initially small sphere is at a distance of r=5 R and slowly move to r=3 R

Electric Potential Energy Between two Charged Particle

$U=k\cdot \frac{q_1q_2}{r^2}$

Initial Potential Energy

$U_1=k\cdot \frac{qq_0}{(5R)^2}$

$U_1=k\cdot \frac{qq_0}{25R^2}$

at r=3R

$U_2=k\cdot \frac{qq_0}{(3R)^2}$

$U_2=k\cdot \frac{qq_0}{9R^2}$

Work Done $=U_2-U_1$

$W=k\cdot \frac{qq_0}{9R^2}-k\cdot \frac{qq_0}{25R^2}$

$W=k\cdot \frac{qq_0}{R^2}\left [ \frac{1}{9}-\frac{1}{25}\right ]$

$W=k\cdot \frac{qq_0}{R^2}\cdot \left [ \frac{16}{225}\right ]$

$W=\frac{1}{4\pi \epsilon _0}\cdot \frac{qq_0}{R^2}\cdot \left [ \frac{16}{225}\right ]$

$W=\frac{4qq_0}{225\pi \epsilon _0R^2}$