Inductance

We have learned about e.m.f., resistance, and capacitance. Let us now investigate inductance. Electrical engineers like to reduce all pieces of electrical apparatus to an equivalent circuit consisting only of e.m.f. sources (e.g., batteries), inductors, capacitors, and resistors. Clearly, once we understand inductors, we shall be ready to apply the laws of electromagnetism to electrical circuits.

Consider two stationary loops of wire, labeled 1 and 2. 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:

$${\mit\Phi}_2 = \int_{\rm loop 2} {\bf B}_1\cdot d{\bf S}_2,$$ (896)

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,

$${\bf B}_1({\bf r}) = \frac{\mu_0 I_1}{4\pi} \oint_{\rm loo... ... \times ({\bf r} - {\bf r}')}{\vert{\bf r} - {\bf r}'\vert^3},$$ (897)

that the magnitude of ${\bf B}_1$ is proportional to the current $I_1$. This is ultimately a consequence of the linearity of Maxwell's equations. Here, $d{\bf l}_1$ is a line element of loop 1 located at position vector ${\bf r}'$. It follows that the flux ${\mit\Phi}_2$ must also be proportional to $I_1$. Thus, we can write

$${\mit\Phi}_2 = M_{21} I_1,$$ (898)

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

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

$${\bf B}_1 = \nabla\times {\bf A}_1.$$ (899)

It follows from Stokes' theorem that

$${\mit\Phi}_2 = \int_{\rm loop 2} {\bf B}_1\cdot d{\bf S}_2... ... d{\bf S}_2 = \oint_{\rm loop 2} {\bf A}_1 \cdot d{\bf l}_2,$$ (900)

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

$${\bf A}_1 ({\bf r}) = \frac{\mu_0 I_1}{4\pi} \oint_{\rm loop 1} \frac{d {\bf l}_1}{\vert{\bf r} - {\bf r}'\vert}.$$ (901)

The above equation is just a special case of the more general law,

$${\bf A}_1({\bf r}) = \frac{\mu_0}{4\pi} \int_{\rm all spac... ...bf j}({\bf r}')} {\vert{\bf r} - {\bf r}'\vert} d^3{\bf r}',$$ (902)

for ${\bf j}({\bf r}') = d{\bf l}_1 I_1/ dl_1 dA$ and $d^3{\bf r}' = dl_1 dA$, where $dA$ is the cross-sectional area of loop 1. Thus,

$${\mit\Phi}_2 = \frac{\mu_0 I_1}{4\pi} \oint_{\rm loop 1... ...frac{d{\bf l}_1\cdot d{\bf l}_2}{\vert{\bf r}- {\bf r}'\vert},$$ (903)

where ${\bf r}$ is now the position vector of the line element $d{\bf l}_2$ of loop 2. The above equation implies that

$$M_{21} = \frac{\mu_0}{4\pi} \oint_{\rm loop 1}\oint_{\rm ... ...frac{d{\bf l}_1\cdot d{\bf l}_2}{\vert{\bf r}- {\bf r}'\vert}.$$ (904)

In fact, mutual inductances are rarely worked out from first principles--it is usually too difficult. However, the above formula tells us two important things. Firstly, the mutual inductance of two 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,

$$M_{21} = M_{12}.$$ (905)

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

We have seen that a current $I$ flowing around some 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

$${\mit\Phi} = L I.$$ (906)

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

Inductance is measured in S.I. units called henries (H): 1 henry is 1 volt-second per ampere. The henry, like the farad, is a rather unwieldy unit, since most real-life inductors have a inductances of order a micro-henry.