Question

1270 A circular loop lying in the plane of the paper has acting through it a magnetic field that makes everywhere an angle a to the normal to the plane of the loop, and does not vary with position but varies with time according to B-kt. The loop has radius a (a) (b) Find the electric field in the plane of the paper and tangential to the loop. Solve the more general problem in which the B field has the same direction as in (a) but varies with both time and distance from the origin according to B krt. Find the E field as a function of time and distance from the origin and angle a

Answer 1

Since direction of magnetic is not specified, from, considering, as shown in the figure Now, we know that closedcontour E^rightarrow dl = partial differential phi_B/partial differential t_(1) Now in the loop as shown in the figure phi_B = integral B^rightarrow n^ds = integral B cos alpha ds = cos alpha B pi r^2 = kt^2 pi r^2 cos alpha__(2) Now consider direction of dl as shown in the figure, as the direction of E as I closedcountour E^rightarrow.dl^rightarrow = E 2 pi r_(3) using (2) and (3) in (1) E 2 pi r = -2 kt pi r^2 cos alpha E = -krt cos alpha r = a E = -kat cos alpha or E = kat cos alpha Note: rightarrow -ve sign shows, direction of E^rightarrow is opposite of what we have considered (b) - Now in second case, for calculating phi_B, in the loop of radius a, we will consider phi_B