The total inductance of two coupled coils in the series opposing and series aiding connections are 12 mH and 38 mH, respectively. Find the mutual inductance between the coils.

Correct option is D

$$\text{Given:} \\[4pt]\bullet \; L_{\text{aiding}} = 38\,\text{mH} \\[4pt]\bullet \; L_{\text{opposing}} = 12\,\text{mH} \\[8pt]\text{For two magnetically coupled coils:} \\[4pt]L_{\text{aiding}} = L_1 + L_2 + 2M \\[4pt]L_{\text{opposing}} = L_1 + L_2 - 2M \\[8pt]\text{Subtracting the two equations:} \\[4pt]L_{\text{aiding}} - L_{\text{opposing}} = 4M \\[8pt]38 - 12 = 4M \Rightarrow 26 = 4M \Rightarrow M = \frac{26}{4} = 6.5\,\text{mH}$$​​