Question

A long uniform cylindrical beam of radius R consists of positively charged particles each of charge q, mass m and velocity V along positive x direction. The axis of the beam is the x-axis. The beam is incident on a region having magnetic field in y-z plane. The magnetic field in the region is confined to 0≤x≤Δx and d to x≤d + 2Δx , (Δx is very small). The magnetic field lines are circular in yz plane as shown. The magnitude of the field is given by B = B₀r where B₀ is a constant and r is distance from the x-axis. [Assume no interaction among charges within the beam]

• A. If d = $\frac{mV}{2qB₀Δx}$, then final emergent beam is parallel
• B. If d = $\frac{mV}{2qB₀Δx}$, then final emergent beam is convergent
• C. If d = $\frac{mV}{qB₀Δx}$, then final emergent beam is convergent
• D. None of these

Answer 1

Answer: (A)

If d = $\frac{mV}{2qB₀Δx}$, then final emergent beam is parallel

Answer 2

When the charged particles pass through the first magnetic field region, they acquire a transverse velocity towards the axis. This causes them to move towards the axis as they travel in the free space. When they enter the second magnetic field region, the force is directed away from the axis, which opposes the inward transverse velocity. For the beam to be parallel after the second region, the net transverse velocity must be zero. The condition $d = \frac{mV}{2qB₀Δx}$ ensures that the outward impulse in the second region exactly cancels the inward impulse from the first region and the inward displacement during the free travel.