📂Electrodynamics

Mutual Inductance

Explanation¹

Consider two fixed conducting loops as shown in the figure above. If a steady current $I_{1}$ flows through loop 1, it generates a magnetic field $\mathbf{B}_{1}$.(Ampère’s law) Some of the magnetic field lines $\mathbf{B}_{1}$ will pass through loop 2. Hence, one can talk about the flux $\Phi_{2}=\mathbf{B}_{1} \cdot d\mathbf{a}_{2}$ of $\mathbf{B}_{1}$ that passes through loop 2. At this point, $d\mathbf{a}_{2}$ is the area vector of loop 2 whose magnitude is equal to the area enclosed by loop 2 and the direction is perpendicular to the plane enclosed by loop 2.

If the shape of loop 1 is not simple, like a circle or a square, calculating $\mathbf{B}_{1}$ in reality is difficult, therefore calculating the flux $\Phi_{2}$ is also challenging. However, by examining the Biot-Savart Law, we can get an important hint.

$$\mathbf{B}_{1} =\dfrac{ \mu_{0}}{4\pi} I_{1} \oint \dfrac{d \mathbf{l} \times \crH}{\cR ^2}$$

That is, the magnetic field $\mathbf{B}_{1}$ is proportional to the current $I_{1}$ flowing in loop 1. Since $\Phi_{2}=\mathbf{B}_{1} \cdot d\mathbf{a}_{2}$, $\Phi_{2}$ is also proportional to $I_{1}$. Thus, if we denote the proportionality constant as $M_{21}$, it can be written as follows.

$$\Phi_{2}=M_{21}I_{1}$$

This proportionality constant is called mutual inductance. Mutual inductance can be obtained by expressing the flux in terms of vector potential and applying Stokes’ theorem to it.

$$\Phi_{2} = \int \mathbf{B}_{1} \cdot d \mathbf{a}_{2} = \int(\nabla \times \mathbf{A}_{1}) \cdot d \mathbf{a}_{2} = \oint \mathbf{A}_{1} \cdot d \mathbf{l}_{2}$$

Since $\displaystyle \mathbf{A}_{1}=\dfrac{\mu_{0} I_{1}}{4 \pi} \oint \dfrac{d \mathbf{l}_{1}}{\cR}$,

$$\Phi_{2} =\dfrac{\mu_{0} I_{1}}{4 \pi} \oint \left( \oint \dfrac{ d\mathbf{l}_{1}}{\cR} \right)\cdot d \mathbf{l}_{2}$$

Thus, the mutual inductance is

$$M_{21}=\dfrac{\mu_{0} }{4 \pi} \oint \left( \oint \dfrac{ d\mathbf{l}_{1} \cdot d \mathbf{l}_{2} }{\cR} \right)$$

This equation is called the Neumann formula. A close look at the Neumann formula reveals that it does not include any terms for the current $I_{1}$, and the line integrals of the two loops are connected through a dot product. Therefore, the following two important facts can be learned:

1. The mutual inductance $M_{21}$ is a purely geometrical quantity and is independent of the current flowing in the loops. It is determined by the distance between the two loops, their size, shape, etc.
2. Since $d \mathbf{l}_{1} \cdot d\mathbf{l}_{2}=d\mathbf{l}_{2} \cdot d\mathbf{l}_{1}$, it implies $M_{21}=M_{12}=M$.

Summarizing these facts, one can conclude,

Regardless of the shapes and positions of the two loops, the flux of the magnetic field $\mathbf{B}_{1}$ produced when a current $I$ flows through loop 1 and passes through loop 2, is equal to the flux of the magnetic field $\mathbf{B}_{2}$ produced when the same current $I$ flows through loop 2 and passes through loop 1.