The current in a primary coil of a step-down transformer having N_p...

The current in the primary coil is given by $i = i_0 \cos\omega t$.

The magnetic flux linked with the secondary coil is $\phi_s = M i$, where $M$ is the mutual inductance between the coils.

According to Faraday's law of electromagnetic induction, the induced emf in the secondary coil is given by:

$e = -\dfrac{d\phi_s}{dt}$

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

Differentiating the expression for current with respect to time $t$:

$\dfrac{di}{dt} = \dfrac{d}{dt}(i_0 \cos\omega t) = -i_0 \omega \sin\omega t$

Substituting this back into the equation for induced emf:

$e = -M (-i_0 \omega \sin\omega t)$

$e = \omega M i_0 \sin\omega t$

Answer: $\omega M i_0 \sin\omega t$