Question

A charged particle moves in a magnetic field with velocity $ v $ perpendicular to the field $ B $. The radius of the circular path is $ r $. Which of the following expressions gives the charge $ q $ of the particle?

• A. $ q = \frac{mv}{Br} $
• B. $ q = \frac{mvB}{r} $
• C. $ q = \frac{mB}{vr} $
• D. $ q = \frac{Bvr}{m} $

Hint:

Use the relationship between the Lorentz force and centripetal force to derive the charge of the particle.

Answer 1

The Correct Option is A

Step 1: Analyze the Motion.
A charged particle moving perpendicular to a magnetic field experiences a Lorentz force, which acts as a centripetal force: $$ F = qvB, $$ where: - $ q $ represents the particle's charge, - $ v $ denotes the particle's velocity, - $ B $ signifies the magnetic field strength. This force facilitates the centripetal acceleration required for circular motion. The centripetal force is defined as: $$ F_{\text{centripetal}} = \frac{mv^2}{r}, $$ where: - $ m $ is the particle's mass, - $ r $ is the radius of the circular trajectory. By equating the Lorentz force and the centripetal force: $$ qvB = \frac{mv^2}{r}. $$ Step 2: Isolate Charge $ q $.
Rearranging the equation to solve for $ q $: $$ q = \frac{mv^2}{Br}. $$ Step 3: Compare with Options.
The provided options are: A) $ q = \frac{mv}{Br} $
B) $ q = \frac{mvB}{r} $
C) $ q = \frac{mB}{vr} $
D) $ q = \frac{Bvr}{m} $. Our derived expression is: $$ q = \frac{mv}{Br}. $$ Consequently, the correct option is: $$ \boxed{\text{A}} $$