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**Tip 1: Finding mutual inductance of two inductors of different shapes.**There is a general equality between mutual inductance of inductor 1 w.r.t inductor 2 ($M_{12}$) and that of inductor 2 w.r.t. inductor 1 ($M_{21}$)

$⟹M_{12}=M_{21}$

So for a given pair of inductors, we can either calculate $M_{12}$ or $M_{21}$ , whichever is easier to calculate. But how do we decide that? We follow the given step-by-step strategy.

- Usually, one of the two inductors (say coil 1) is dimensionally much much smaller than the other inductor (say coil 2). In such cases, we can assume the magnetic field by the large inductor to be uniform throughout the area/ volume of the small inductor. Thus it becomes much easier to write the flux linkage for the small inductor due to the large inductor.
- We write the flux linkage of coil 1 due to current in coil 2,$N_{1}ϕ_{1}=M_{1}2I_{2}$
- Following are some of the examples in which we will find calculating $M_{12}$ much easier than $M_{21}$

Here the inner solenoid is effectively immersed in a uniform magnetic field due to the outer solenoid. Thus, the calculation of $M_{12}$ would be easy. However, it would be extremely difficult to calculate the flux linkage with the outer solenoid as the magnetic field due to the inner solenoid would vary across the length as well as cross-section of the outer solenoid.

Similarly, we can easily find $M_{12}$ here when $l<<<<L$ as we have to find magnetic field at the centre of the outer square loop and consider it to be uniform throughout the area of the inner square loop.

Same approach can be used for this situation where we can write $M_{21}$ easily by writing the magnetic field due to ring 1 on its axis.

Find an expression of mutual inductance for shown concentric co-planner circular and regular hexagonal loops $(a>>r):$

A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Figure . What is the mutual inductance M for the loopwire combination if N = 100, a = 1.0 cm, b = 8.0 cm, and l = 30 cm?