Problem Statement: A small ball of mass m and charge q which is tied with an insulating string of length L is rotating on a vertical circular path under gravity in a uniform electric field E. The tension in the string is minimum for (a)theta=tan^-1 (qE/mg) (b) pi tan^-1 (qE/mg)

Solution:

Understanding the problem:
The problem describes a small ball of mass m and charge q that is rotating in a vertical circular path under the influence of gravity and a uniform electric field E. The ball is tied with an insulating string of length L. We need to determine the angle theta at which the tension in the string is minimum.

Analysis:
To analyze the problem, we need to consider the forces acting on the ball. These are:
- Gravitational force: This is given by Fg = mg, where m is the mass of the ball and g is the acceleration due to gravity.
- Electric force: This is given by Fe = qE, where q is the charge on the ball and E is the electric field.
- Tension in the string: This is given by T.

The ball is moving in a vertical circular path, so we can resolve the gravitational force into two components:
- Tangential force: This is given by Ft = mg sin(theta), where theta is the angle between the tangent to the circular path and the vertical.
- Radial force: This is given by Fr = mg cos(theta).

Since the ball is in equilibrium, the sum of the forces in the radial direction must be zero. Therefore, we have:
T - Fr - Fe = 0
or, T - mg cos(theta) - qE = 0

Similarly, the sum of the forces in the tangential direction must be zero. Therefore, we have:
Ft = ma
or, mg sin(theta) = ma

Since the ball is moving in a circular path, the acceleration is given by a = v^2/R, where v is the speed of the ball and R is the radius of the circular path. Therefore, we have:
mg sin(theta) = mv^2/R
or, v^2 = gR sin(theta)

Substituting this value of v^2 in the expression for the tension, we get:
T = m(g cos(theta) + qE)/sin(theta)

To find the angle theta at which the tension is minimum, we need to differentiate this expression with respect to theta and equate it to zero. Solving this equation, we get:
tan(theta) = qE/mg
or, theta = tan^-1(qE/mg)

Therefore, the correct answer is (a) theta = tan^-1(qE/mg).