Question

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.

Answer 1

The surface charge density of the plate is 7.5$\times$10−7 C/m2.

Explanation:

Please note that there are two forces acting on the ball.  

They are :-

• The weight of the ball, W = mg
• The coulomb force acting on the charged ball due to the electric field of the plate, F = qE
• As a result of these forces, a tension develops in the thread.
• Say, the surface charge density on the plate be σ.

Electric field of a plate,

E = σ2∈0σ2∈0

• It is provided that in equilibrium, the thread makes an angle of 60° with the vertical.
• Once we resolve the tension in the string along horizontal and vertical directions, we get:
• T cos 60°=mgT sin 60°= qE⇒tan 60°=qEmg⇒ E=mg tan 60°qT cos 60°=mgT sin 60°= qE⇒tan 60°=qEmg⇒ E=mg tan 60°q

Also, electric field due to a plate,

E=σ2 ∈0=mg tan60°q  

σ=2 ∈0 mg tan 60°q  

$\begin{array}{l}{\sigma=2 \times(8.85 \times 10-12) \times(10 \times 10-3 \times 9.8) \times 1.7324 .0 \times 10-6} \\{\sigma=7.5 \times 10-7 \mathrm{C} / \mathrm{m} 2}\end{array}$