Two pith balls each of mass 2 mg are suspended by very light thread of 20 cm long from a common pt. each ball has the same charge so that the threads form angles of 30° with the vertical. what is the charge on each ball?

To find the charge on each ball, we can use the concept of electrostatics and the principles of forces acting on charged objects.

First, let's define the variables given in the problem:
- The mass of each pith ball: m = 2 mg = 2 × 10^(-6) kg
- The length of the thread: L = 20 cm = 0.2 m
- The angle formed by the thread with the vertical: θ = 30°

Now, let's examine the forces acting on each pith ball:

1. Gravitational Force (Weight):
The gravitational force acting on each pith ball is given by the formula Fg = mg, where g is the acceleration due to gravity. In this case, we assume g ≈ 9.8 m/s^2.

Fg = (2 × 10^(-6) kg) × (9.8 m/s^2) = 1.96 × 10^(-5) N

2. Electrostatic Force:
The electrostatic force acting on each pith ball is given by the formula Fe = (kQ^2)/r^2, where k is the Coulomb constant (k ≈ 9 × 10^9 N m^2/C^2), Q is the charge on each ball, and r is the length of the thread.

Since the threads form angles of 30° with the vertical, we need to analyze the forces in the vertical and horizontal directions separately.

In the horizontal direction, the electrostatic forces on the two balls cancel out, since they have the same charge and the strings are at the same angle.

In the vertical direction, the net force is balanced by the tension in the thread:
Tcos(30°) = 2 × Fg

The tension in the thread is related to the electrostatic force by:
Tsin(30°) = 2 × Fe

Since the masses of the pith balls are very small, we can assume that the only significant forces in the vertical direction are the electrostatic forces and tensions in the thread.

Dividing the two equations, we get:
Tsin(30°) / Tcos(30°) = 2 × Fe / 2 × Fg
tan(30°) = Fe / Fg

tan(30°) = (kQ^2)/(r^2 × mg)

Now, we can solve for Q:

Q^2 = (r^2 × mg × tan(30°)) / k

Q = √[(r^2 × mg × tan(30°)) / k]

Substituting the given values:
Q = √[(0.2^2 m × 2 × 10^(-6) kg × tan(30°)) / (9 × 10^9 N m^2/C^2)]

Calculating this expression gives us the charge on each pith ball.