Q. 19

# One end of a 10 cm long silk thread is fixed to a large vertical surface of a charged nonconducting plate and the other end is fastened to a small ball having a mass of 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.

Given:

Mass of ball=10g

Charge of ball=4.0× 10-6C

Equilibrium angle of thread with vertical=60°

Let the surface charge density on the plate be σ

The forces acting on the ball are

• Weight of the ball downwards (W=mg)

Where m=mass of ball

g=acceleration due to gravity

• Electric force due to non-conducting plate producing electric field E (F=qE)

Where q=charge of the ball

E=electric field intensity

• Tension force (T) due to string at an angle of 60° from vertical

Electric field due to a thin non-conducting plate of surface charge density σ is given by

……(i)

The tension force T due to string is divided into horizontal and vertical components given by Tsin60° and Tcos60°

Since the ball is in equilibrium, the net horizontal and vertical force on the ball is zero

Applying equilibrium along horizontal direction, we get

…(ii)

Similarly, applying equilibrium along vertical direction, we get

….(iii)

Dividing eqn. (ii) by (iii)

Putting the value of E from eqn.(i), we get,

Putting the values of ϵ0,m,g and q in the above equation

C/m2

Therefore the surface charge density on the plate is given by 7.5× 10-7C/m2

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