Electric charges and field Part - 4

1. Concept of electrostatic force:

1. Force on a point charge due to uniformly charged rod: A rod of length L is uniformly charged by Q placed at a distance D from a point charge q at P. Charge per unit length of the rod is λ = . We consider an elementary part of the rod of length dx at a distance x from P. The charge of dx is dQ = dx.

The electric field at P due to dQ is dE = k()dx.

So, the total electric field at P due to the entire rod is

E = dE = = = =

The force on charge q is F = qE = .

2. Tension on a charged ring: A circular ring of radius r is uniformly charged by q, so the charge per unit length is λ = . AB is the elementary part of the ring of length dl with charge dq = λdl = . A point charge Q is placed at the centre O of the ring.

The force on AB due to Q is dF = = .

Let T is the increase in tension on the ring to balance the repulsive force. The inward tension for the part AB is 2Tsin .

As θ is very small then, from figure = or, dl = rθ.

Then, 2Tsin =

Or, 2T =

Or, T =

T = .

3. Increased radius of the ring due to excess tension: Let us consider r is the increased in radius of the ring due to excess tension. So, increased in length of the ring is (r + r) – 2πr = 2π∆r.

Therefore strain = = . Stress = where A is the cross-sectional area of the ring.

If Y is the Young’s modulus of the material of the ring then, Y = =

Or, r = = [F = ].

4. Vertical circular motion of a charged ball: A ball of mass m charge q is tied at one end of an insulated thread of length L. The other end of the thread is connected with a fixed charge -Q. The minimum horizontal speed is required at the top point of the vertical circular motion of the ball is u. The electrostatic force between two charges is F = .