A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant rate dB/dt. Consider a circle of radius r coaxial with the cylindrical region.
(a) Find the magnitude of the electric field E at a point on the circumference of the circle.
(b) Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude forum in part (a).
Given:
Magnetic field = B
Rate of increase of magnetic field = dB/dt
Radius = r
Formula used:
(a) Induced emf 
… (i), where 
= magnetic flux, t = time
Now,
= B.A where B = magnetic field, A = area
Hence, 
… (ii)
For the circular loop, 
… (iii), where A = area, r = radius
Let the electric field be E
Hence, 
… (iv) ,where dr = element of length, E’ = emf
Hence, for this loop, 
, where r = radius
⇒ 
⇒ 
(Ans)
(b) When the square is considered, A = (2r)2 = 4r2, where A = area, r = radius
In this case, 
(perimeter of square)
Hence, from 
, where E = electric field, dr = length element, E’ = emf, we get
=> electric field 
(Ans)