Question

A small sphere with mass m carries a positive charge q and is attached to one end of a silk fiber of length L. The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density σ. Assume that the sphere is in equilibrium and find the angle that fiber makes with the vertical sheet. Express your answer in terms of the variables q, σ, m, and appropriate constants.

Answer 1

Here in the given situation there is a constant electric field due to large sheet

This constant electric field is given as

$E = \frac{\sigma}{2\epsilon_0}$

now we know that force on the ball due to this electric field will be in horizontal direction given as

$F = qE$

$F = \frac{\sigma q}{2\epsilon_0}$

now this horizontal force will be balanced by horizontal component of the tension in the string

$Tsin\theta = \frac{\sigma q}{2 \epsilon_0}$

now similarly we can say that vertical component of tension force in the string will balance the weight of small sphere

$Tcos\theta = mg$

now from above two equations we will have

$tan\theta = \frac{\sigma q}{2\epsilon_0 mg}$

$\theta = tan^{-1}\frac{\sigma q}{2\epsilon_0 mg}$