Explanation

(d) When the two coils are joined in series such that the winding of one is opposite to the other, then the emf produced in first coil is 180${^\circ}$ out of phase of the emf produced in second coil. Thus, emf produced in first coil is negative and the emf produced in second coil is positive, so net inductance is
$L = L_{1} + L_{2} = L + L$
From Faraday’s law of electromagnetic induction
$\phi = Li,$ where $\phi$ is flux and $i$ the current
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L = - \frac{\phi}{i} + \frac{\phi}{i}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L$=0