Electron Path Radius in Magnetic Field

Electron Path Radius vs Speed

Question: When the speed of an electron in a perpendicular magnetic field is doubled, the radius of its circular path becomes:

(a) 2.0 cm

(b) 4.0 cm

(c) 0.5 cm

(d) 1.0 cm

Solution

Step 1: Radius Formula
Magnetic force provides centripetal force: \[ qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB} \]

Step 2: Proportionality
Radius \( r \propto v \) (all other quantities constant).
Original radius = 2 cm at speed \( v \).
New speed = \( 2v \implies \) New radius = \( 2 \times 2 \, \text{cm} = 4 \, \text{cm} \).

Answer: Option (b) \( 4.0 \, \text{cm} \) is correct.