Two coils P and S have a mutual inductance of 5 × 10^ -3 \ H . If...

The induced e.m.f. in the secondary coil is given by:

$e = -M \dfrac{dI}{dt}$

Given $M = 5 \times 10^{-3}\ \text{H}$ and $I = 10\sin(100\pi t)$.

Differentiating the current with respect to time:

$\dfrac{dI}{dt} = \dfrac{d}{dt} (10\sin(100\pi t)) = 1000\pi \cos(100\pi t)$

Substituting the values into the e.m.f. equation:

$e = -(5 \times 10^{-3}) \times 1000\pi \cos(100\pi t)$

$e = -5\pi \cos(100\pi t)$

The maximum value of the induced e.m.f. corresponds to the maximum magnitude of the cosine function, which is $1$.

e_\max = 5\pi\ \text{V}

Answer: $5\pi$