Two coils have a mutual inductance of $$0.005\,H.$$  The current changes in the first coil according to equation $$i = {i_0}\sin \omega t,{i_0} = 10\,A$$     and $$\omega = 100\,\pi \,rad/s.$$    The maximum value of emf in the second coil is

Solution :
Problem Solving Strategy
Differentiate the given equation of current changing in first coil and find out the maximum change in $$\frac{{di}}{{dt}}.$$
The given equation of current changing in the first coil is $$i = {i_0}\sin \omega t\,......\left( {\text{i}} \right)$$
Differentiating Eq. (i) w.r.t. $$t,$$ we have
$$\eqalign{ & \frac{{di}}{{dt}} = \frac{d}{{dt}}\left( {{i_0}\sin \omega t} \right) \cr & {\text{or}}\,\,\frac{{di}}{{dt}} = {i_0}\frac{d}{{dt}}\left( {\sin \omega t} \right) \cr & {\text{or}}\,\,\frac{{di}}{{dt}} = {i_0}\omega \cos \omega t \cr} $$
For maximum $$\frac{{di}}{{dt}},$$  the value of $$\cos \omega t$$  should be equal to 1.
$${\text{So,}}\,\,{\left( {\frac{{di}}{{dt}}} \right)_{\max }} = {i_0}\omega $$
The maximum value of emf is given by
$$\eqalign{ & \therefore {e_{\max }} = M{\left( {\frac{{di}}{{dt}}} \right)_{\max }} = M{i_0}\omega \cr & {\text{As,}}\,\,M = 0.005H,{i_0} = 10\;A,\omega = 100\,\pi \,rad/s \cr & \therefore {e_{\max }} = 0.005 \times 10 \times 100\pi = 5\pi \cr} $$