a point charge -q (of mass m) is revolving in a circle of rdius r ,around a long positive line of charge density u,as shown .its angular velocity is

To determine the angular velocity of a point charge -q revolving around a long positive line of charge with linear charge density λ, we can analyze the forces acting on the charge and apply some principles of circular motion and electrostatics.

Understanding the Forces at Play

When the point charge -q revolves around the line of charge, it experiences a centripetal force that keeps it in circular motion. This centripetal force is provided by the electric field created by the line charge. Let's break this down step by step.

Electric Field Due to the Line Charge

The electric field (E) created by an infinite line of charge with linear charge density λ at a distance r from the line is given by the formula:

E = (λ / (2πε₀r))

Here, ε₀ is the permittivity of free space. The direction of the electric field is radially outward from the line charge since it is positive.

Force on the Point Charge

The force (F) acting on the point charge -q due to this electric field can be calculated using Coulomb's law:

F = qE

Substituting the expression for the electric field, we get:

F = -q(λ / (2πε₀r))

Since the charge is negative, the force will be directed towards the line charge, which is consistent with the centripetal force required for circular motion.

Centripetal Force Requirement

For an object moving in a circle of radius r with mass m and angular velocity ω, the required centripetal force (F_c) is given by:

F_c = mω²r

Setting Forces Equal

For the point charge to maintain its circular motion, the electric force must equal the centripetal force:

mω²r = -q(λ / (2πε₀r))

Since we are interested in the magnitude, we can drop the negative sign:

mω²r = q(λ / (2πε₀r))

Solving for Angular Velocity

Now, we can solve for the angular velocity ω:

ω² = (qλ) / (2πε₀mr²)

Taking the square root of both sides gives us:

ω = √((qλ) / (2πε₀mr²))

Final Thoughts

This formula provides the angular velocity of the point charge -q revolving around the line of charge. It shows how the charge's mass, the charge density of the line, and the distance from the line all influence the angular velocity. This relationship is crucial in understanding the dynamics of charged particles in electric fields and can be applied in various fields, including physics and engineering.