Electron Frequency in Magnetic Field

Electron in Magnetic Field

Question: An electron (mass \( m \), kinetic energy \( E \)) enters a uniform magnetic field \( B \) perpendicularly. Its frequency is:

(a) \( \frac{eE}{qvB} \)

(b) \( \frac{eB}{2\pi m} \)

(c) \( \frac{2\pi m}{eB} \)

(d) \( \frac{2m}{eBE} \)

Solution

Step 1: Magnetic force provides centripetal force
\( qvB = \frac{mv^2}{r} \) ⇒ \( r = \frac{mv}{qB} \)

Step 2: Relationship between velocity and frequency
\( v = \omega r = 2\pi f r \)

Step 3: Substitute \( r \) from Step 1 into Step 2
\( v = 2\pi f \left(\frac{mv}{qB}\right) \) ⇒ \( f = \frac{qB}{2\pi m} \)

Step 4: For electron (\( q = e \))
\( f = \frac{eB}{2\pi m} \)

Answer: Option (b) is correct