In Exercise 4.11 obtain the frequency of revolution of the electron in its circular orbit. Does the answer depend on the speed of the electron? Explain.

Magnetic field strength, B =6.5 x 10^-4 T

Charge of the electron, e = 1.6 x 10^-19 C

Mass of the electron, m_e = 9.1 x 10^-31 kg

Velocity of the electron, v = 4.8 x 10⁶ m/s

Radius of the orbit, r = 4.2 cm = 0.042 m

Frequency of revolution of the electron = v

Angular frequency of the electron = ω = 2πv

Velocity of the electron is related to the angular frequency as: v = rω

In the circular orbit, the magnetic force on the electron is balanced by the centripetal force.

Hence, we can write:

mv²/r = evB

=> eB = mv/r =m (rω)/r = m (r.2πv)/r

=> v = Be/2πm

This expression for frequency is independent of the speed of the electron.

On substituting the known values in this expression, we get the frequency as

v = ( 6.5 x 10^-4 x  1.6 x 10^-19 )/( 2 x 3.14 x 9.1 x 10^-31

= 1.82  x 10⁶ Hz or approx. 18MHz

Hence ,the frequency of the electron is around 18MHz and is independent of the speed of the electron.