Inductance

We have already learned about the concepts of voltage, resistance, and capacitance. Let us now investigate the concept of inductance. Electrical engineers like to reduce all pieces of electrical circuitry to an equivalent circuit consisting of pure voltage sources, pure inductors, pure capacitors, and pure resistors. Hence, once we understand inductors, we shall be ready to apply the laws of electromagnetism to general electrical circuits.

Consider two stationary loops of wire, labeled 1 and 2. See Figure 2.26. Let us run a steady current $I_1$ around the first loop to produce a magnetic field ${\bf B}_1$. Some of the field-lines of ${\bf B}_1$ will pass through the second loop. Let ${\mit\Phi}_2$ be the flux of ${\bf B}_1$ through loop 2,

$${\mit\Phi}_2 = \int_{\rm loop\, 2} {\bf B}_1\cdot d{\bf S}_2,$$ (2.311)

where $d{\bf S}_2$ is a surface element of loop 2. This flux is generally quite difficult to calculate exactly (unless the two loops have a particularly simple geometry). However, we can infer from the Biot-Savart law [see Equation (2.226)],

$${\bf B}_1({\bf r}) = \frac{\mu_0 \,I_1}{4\pi} \oint_{\rm loop \,1... ...rac{ d{\bf r}_1 \times ({\bf r} - {\bf r}_1)}{\vert{\bf r} - {\bf r}_1\vert^3},$$ (2.312)

that the magnitude of ${\bf B}_1$ is proportional to the current $I_1$. Here, $d{\bf r}_1$ is a line element of loop 1 located at displacement ${\bf r}_1$. It follows that the flux ${\mit\Phi}_2$ must also be proportional to $I_1$. Thus, we can write

$${\mit\Phi}_2 = M_{21}\,I_1,$$ (2.313)

where $M_{21}$ is a constant of proportionality. This constant is termed the mutual inductance of the two loops.

Figure 2.26: Two current-carrying loops.

Let us write the magnetic field ${\bf B}_1$ in terms of a vector potential ${\bf A}_1$, so that

$${\bf B}_1 = \nabla\times {\bf A}_1.$$ (2.314)

It follows from the curl theorem (see Section A.22) that

$${\mit\Phi}_2 = \int_{\rm loop \,2} {\bf B}_1\cdot d{\bf S}_2 = \i... ...s {\bf A}_1 \cdot d{\bf S}_2 = \oint_{\rm loop\, 2} {\bf A}_1 \cdot d{\bf r}_2,$$ (2.315)

where $d{\bf r}_2$ is a line element of loop 2. However, we know that

$${\bf A}_1 ({\bf r}) = \frac{\mu_0 \,I_1}{4\pi} \oint_{\rm loop \,1} \frac{d {\bf r}_1}{\vert{\bf r} - {\bf r}_1\vert}.$$ (2.316)

The previous equation is just a special case of the more general result [see Equation (2.252)],

$${\bf A}_1({\bf r}) = \frac{\mu_0}{4\pi} \int \frac{{\bf j}({\bf r}')} {\vert{\bf r} - {\bf r}'\vert} \,dV',$$ (2.317)

for ${\bf j}({\bf r}_1) = d{\bf r}_1 \,I_1/( dr_1\, dA)$ and $dV' = dr_1 \,dA$, where $dA$ is the cross-sectional area of loop 1. Thus,

$${\mit\Phi}_2 = \frac{\mu_0 \,I_1}{4\pi} \oint_{\rm loop \,1}\oint_{\rm loop \,2} \frac{d{\bf r}_1\cdot d{\bf r}_2}{\vert{\bf r}_2- {\bf r}_1\vert},$$ (2.318)

where ${\bf r}_2$ is the position vector of the line element $d{\bf r}_2$ of loop 2, which implies that

$$M_{21} = \frac{\mu_0}{4\pi} \oint_{\rm loop\, 1}\oint_{\rm loop \,2} \frac{d{\bf r}_1\cdot d{\bf r}_2}{\vert{\bf r}_2- {\bf r}_1\vert}.$$ (2.319)

In fact, mutual inductances are rarely worked out using the previous formula, because it is usually far too difficult. However, this formula, which is known as the Neumann formula, tells us two important things. Firstly, the mutual inductance of two current loops is a purely geometric quantity, having to do with the sizes, shapes, and relative orientations of the loops. Secondly, the integral is unchanged if we switch the roles of loops 1 and 2. In other words,

$$M_{21} = M_{12}.$$ (2.320)

Hence, we can drop the subscripts, and just call both of these quantities $M$. This result implies that no matter what the shapes and relative positions of the two loops, the magnetic flux through loop 2 when a current $I$ runs around loop 1 is exactly the same as the flux through loop 1 when the same current runs around loop 2.

We have seen that a current $I$ flowing around some wire loop, 1, generates a magnetic flux linking some other loop, 2. However, flux is also generated through the first loop. As before, the magnetic field, and, therefore, the flux, ${\mit\Phi}$, is proportional to the current, so we can write

$${\mit\Phi} = L\, I.$$ (2.321)

The constant of proportionality $L$ is called the self inductance. Like $M$ it only depends on the geometry of the loop.

The SI unit of inductance is the henry (H), which is equivalent to a volt-second per ampere. The henry, like the farad, is a rather unwieldy unit, because inductors in electrical circuits typically have a inductances of order a micro-henry.