An electron is moving in a circular orbit of radius R . . .

Question : An electron is moving in a circular orbit of radius R with an angular acceleration α. At the centre of the orbit is kept a conducting loop of radius r, (r<<R). The emf induced in the smaller loop due to the motion of the electron is :

a) Zero

b) µ₀er²α/4R

c) µ₀er²α/4πR

d) none of these.

Doubt by Jayant

Solution : 

We know, According to Faraday's laws of EMI, the magnitude of emf induced is given by 

ε=dΦ/dt — (1)

Φ=BAcos0°
Φ=BA — (2)

Magnetic field at the centre due to rotation of electron 
B = µ₀I/2R — (3)

Also,
I =q/t
I = e/T
I = e/(2π/ω)
I = eω/2π

Putting in equation (3)
B = µ₀(eω/2π)/2R
B = µ₀eω/4πR

Substituting in equation (2)

Φ=BA 
Φ=[µ₀eω/4πR][πr²]
Φ=µ₀eωr²/4R

Substituting this is equation (1)

ε=dΦ/dt 

ε=d[µ₀eωr²/4R]/dt 
ε=µ₀er²/4R [dω/dt]
ε=µ₀er²/4R (α) [∵dω/dt = α]
ε=µ₀er²α/4R

Hence, b) is the correct option.