Q64 of 104 Page 306

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)


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