Charged Particle in Magnetic Field

Charged Particle in Magnetic Field

Question: A positively charged particle moving east enters a region with a uniform magnetic field directed vertically upwards. The particle will:

(a) Move in a circular path with decreased speed

(b) Move in a circular path with uniform speed

(c) Get deflected vertically upward

(d) Move in a circular path with increased speed

Solution

Step 1: Magnetic force direction (Right-Hand Rule)
For a positive charge moving east (\( \vec{v} \)) in an upward magnetic field (\( \vec{B} \)):
\( \vec{F} = q(\vec{v} \times \vec{B}) \).
Using the right-hand rule: Force (\( \vec{F} \)) acts southward (perpendicular to both velocity and field).

Step 2: Nature of motion
The magnetic force provides centripetal force:
\( \frac{mv^2}{r} = qvB \Rightarrow r = \frac{mv}{qB} \).
The particle follows a circular path in the horizontal plane.

Step 3: Speed analysis
Magnetic force does zero work (\( \vec{F} \perp \vec{v} \)), so kinetic energy and speed remain constant.

Answer: Option (b) is correct. The particle moves in a circular path with uniform speed.