Question

Two spherical, non conducting, and very thin shells of uniformly distributed positive charge $Q$ and radius $d$ are located a distance $10d$ from each other. A positive point charge $q$ is placed inside one of the shells at a distance $d/2$ from the center, on the line connecting the centers of the two shells, as show in the figure. What is the net force on the charge $q$?

• A. $\frac{Qq}{361\pi {\epsilon }_0{d}^2}$ to the left
• B. $\frac{Qq}{361\pi {\epsilon }_0{d}^2}$ to the right
• C. $\frac{362Qq}{361\pi {\epsilon }_0{d}^2}$ to the left
• D. $\frac{360Qq}{361\pi {\epsilon }_0{d}^2}$ to the right

Answer 1

The correct option is A $\frac{Qq}{361\pi {\epsilon }_0{d}^2}$ to the left

The charge is inside one of the shells. The electric field due to this shell inside is zero, so no force would act on it due to this shell. The only force experienced by the charge would be due to the electric field due to the shell which is at the distance.

Thus we get

$E=\frac{kQ}{r^2}$

where, $r=10d-\frac{d}{2}=\frac{19d}{2}$

Therefore,

$E=\frac{kQ}{(\frac{19d}{2}{)}^2}$

$E=\frac{4kQ}{361d^2}$

$E=\frac{Q}{361\pi {ϵ}_0{d}^2}$

Now, considering the force exerted by opposite sphere to the left, we can write

$F=qE=\frac{qQ}{361\pi {ϵ}_0{d}^2}$