A circular coil of fixed radius r lies in a region where the magnetic field is uniform and directed normal to the plane of the coil. The field magnitude varies with time according to
B(t)=B0(3t−t2)(0≤t≤3s),
where B0 is a positive constant. Simultaneously, the coil is fitted with a mechanism that decreases its area according to
A(t)=A0(1−kt),
where A0 and k are positive constants chosen so that A(t) remains positive for the entire interval 0≤t≤3s. The plane of the coil does not rotate.
(i) Write an expression for the magnetic flux Φ(t) linked with the coil as a function of time.
(ii) Obtain an expression for the induced emf E(t) and simplify it as far as possible.
(iii) Determine the exact instant(s) in the interval 0≤t≤3s at which the induced emf is zero.
(iv) Without performing any additional algebra, state during which sub-intervals of time (0≤t≤3s) the induced current is clockwise and during which it is counter-clockwise. Base your answer on the signs of the individual contributions you obtained in part (ii).