Question

A charged particle having charge $q$ is moving perpendicularly to the uniform magnetic field with linear speed $v$ on a circular path of radius $R$. The periodic time of revolution of a particle ______.

• A. depends on $v$ but does not depend on $R$.
• B. do not depend on $v$ and $R$ both.
• C. depends on $R$ but does not depend on $v$.
• D. depend on $v$ and $R$ both.

Hint:

This principle is why Cyclotrons work! Even as the particle gains speed and moves in a larger circle, the time it takes to complete one lap stays exactly the same.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When a charged particle moves perpendicularly to a magnetic field, the magnetic force ($qvB$) provides the necessary centripetal force ($mv^2/R$).
Step 2: Formula Derivation:
Equating the forces: $qvB = \frac{mv^2}{R} \implies v = \frac{qBR}{m}$.
The time period $T$ is the circumference divided by speed: $$T = \frac{2\pi R}{v} = \frac{2\pi R}{(qBR/m)} = \frac{2\pi m}{qB}$$ The resulting formula for $T$ contains only mass, charge, and magnetic field strength.
Step 3: Final Answer:
The time period is independent of both speed ($v$) and radius ($R$).