# Polarity of Mutual Induction

According to the question voltage across the inductor in left circuit is given by

$$V_1 = L\frac{di_1}{dt} \pm M\frac{di_2}{dt}$$ Where $$i_1$$ is the current in left circuit and $$i_2$$ is the current in the right circuit.

Similarly voltage across the inductor in the right circuit is given by

$$V_2 = L\frac{di_2}{dt} \pm M\frac{di_1}{dt}$$

Now, how do I find whether the mutually induced emf is added or subtracted?

Question is taken from JEE Advanced 2020, they have given the answer by assuming that mutually induced emf is positive.

In your case, the polarities are ambiguous. It sounds like they just want to take the currents as being defined in the diagram and take $$M > 0$$, but know that this isn't normally a given and ideally in the real world you would be given more information.
• @MMS05 Even if the inductors are placed next to each other and the currents are flowing in opposite directions (up and down), the sign of the mutual inductance also depends on the way the coils are wound (i.e. clockwise or counterclockwise as seen from above). If they are both wound the same way, yes, $M > 0$ if the currents are in opposite directions (as defined in your diagram) and $M < 0$ if they are in the same direction.
By ancient convention lost in the fog of history $$M$$ itself is a signed quantity, so here it is to be taken with a positive sign.