One end of a 10 cm long silk thread is fixed to a large vertical surface of a charged non-conducting plate and the other end is fastened to a small ball of mass 10 g and a charge of 4.0× 10^{-6} C. In equilibrium, the thread makes an angle of 60° with the vertical. Find the surface charge density on the plate.

#### Solution

There are two forces acting on the ball. These are

(1) Weight of the ball, W = mg

(2) Coulomb force acting on the charged ball due to the electric field of the plate, F = qE

Due to these forces,a tension develops in the thread.

Let the surface charge density on the plate be σ.

Electric field of a plate,

`"E" = sigma /(2∈_0)`

It is given that in equilibrium, the thread makes an angle of 60° with the vertical.

Resolving the tension in the string along horizontal and vertical directions, we get:

T cos 60° = mg

T sin 60° = qE

`=> tan 60° = "qE"/"mg"`

`=> "E" = ("mg" tan 60 °)/q`

Also, electric field due to a plate,

`"E" = sigma/ 2 ∈_0 =( "mg" tan 60°)/q`

`sigma =(2 ∈_0 "mg" tan 60 °)/"q"`

`sigma =( 2 xx (8.55 xx 10^-12 ) xx ( 10 xx 10 ^-3 xx 9.8) xx 1.7320)/(4.0 xx 10^-6)`

σ = 7.5× 10^{-7} C/m^{2}