Two long concentric conducting cylinders with cross section have radii R and 2R. The inner one carries a charge l per unit length. A particle of mass m and negative charge - q is given velocity v0 perpendicular to axis of cylinders and perpendicular to radial direction at a distance 3R/2 from the common axis of cylinders.We have to find the velocity for which particle moves in circular path .

To determine the velocity at which a negatively charged particle moves in a circular path between two concentric conducting cylinders, we need to analyze the electric field created by the charged inner cylinder and the forces acting on the particle. Let's break this down step by step.

Understanding the Electric Field

The inner cylinder has a linear charge density of λ (charge per unit length). The electric field (E) generated by an infinitely long charged cylinder at a distance r from its axis is given by the formula:

E = (λ / (2πε₀r))

Here, ε₀ is the permittivity of free space. Since the particle is located at a distance of 3R/2 from the axis of the cylinders, we can substitute r = 3R/2 into the equation:

E = (λ / (2πε₀(3R/2))) = (λ / (3πε₀R))

Force on the Charged Particle

The negatively charged particle experiences a force due to the electric field. The force (F) acting on the particle can be calculated using the equation:

F = qE

Substituting the expression for the electric field, we get:

F = -q(λ / (3πε₀R))

The negative sign indicates that the force is directed opposite to the electric field, which is expected for a negatively charged particle.

Circular Motion Condition

For the particle to move in a circular path, the centripetal force required to keep it in that path must equal the electric force acting on it. The centripetal force (F_c) can be expressed as:

F_c = (mv²) / r

In this case, r is the distance from the axis of the cylinders, which is 3R/2. Therefore, we can rewrite the centripetal force as:

F_c = (mv²) / (3R/2) = (2mv²) / (3R)

Setting Forces Equal

To find the velocity v required for circular motion, we set the centripetal force equal to the electric force:

(2mv²) / (3R) = q(λ / (3πε₀R))

Now, we can simplify this equation. Multiplying both sides by 3R gives:

2mv² = q(λ / (ε₀))

Solving for Velocity

Now, we can isolate v²:

v² = (qλ) / (2mε₀)

Taking the square root of both sides, we find the velocity v:

v = √((qλ) / (2mε₀))

Final Thoughts

This expression gives us the velocity at which the negatively charged particle must move to maintain a circular path between the two concentric cylinders. The key takeaway is that the balance between the electric force due to the inner cylinder and the required centripetal force for circular motion determines this velocity. By understanding the interplay of electric fields and forces, we can analyze the motion of charged particles in electric fields effectively.