Problem: Three point charges of charge q each are moving along a circle of radius R and a point charge of -2q is placed at the centre of the circle. If charges are revolving with constant and same speed in the circle then calculate the speed of charges.

Solution:

Step 1: Understanding the problem

The problem involves four point charges: three positive charges, each with charge q, and one negative charge with charge -2q. The charges are moving along a circle of radius R with constant and equal speeds.

Step 2: Finding the force between the charges

The positive charges will repel each other, while the negative charge will attract the positive charges towards the centre of the circle. The force between two charges can be calculated using Coulomb's law:

F = k(q_1q_2)/r^2

Where F is the force between the charges, k is Coulomb's constant, q_1 and q_2 are the charges, and r is the distance between the charges.

Step 3: Calculating the net force on each charge

The net force on each of the positive charges will be towards the centre of the circle, due to the attraction from the negative charge. The net force can be calculated by adding up the individual forces on each charge. Since the charges are equidistant from the centre of the circle, the net force on each charge will be the same.

Step 4: Applying Newton's second law

The net force on each charge will cause it to accelerate towards the centre of the circle. The acceleration can be calculated using Newton's second law:

F = ma

Where F is the net force on the charge, m is its mass, and a is its acceleration.

Step 5: Calculating the speed of the charges

Since the charges are moving with constant speed, the acceleration towards the centre of the circle must be balanced by a centrifugal force, given by:

F = mv^2/r

Where v is the speed of the charge, and r is the radius of the circle.

Setting the two expressions for F equal to each other, we get:

k(q^2)/r^2 = mv^2/r

Solving for v, we get:

v = sqrt(kq^2/2mr)

Step 6: Plugging in the values

Plugging in the values for k, q, m, and r, we get:

v = sqrt((9x10^9 Nm^2/C^2)(q^2)/(2(4/3πR^3)(3q)))

Simplifying, we get:

v = sqrt((27π/16)(kq/R))

Therefore, the speed of the charges is given by:

v = sqrt((27π/16)(kq/R))