Up: Magnetic induction
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
around the first loop to produce a magnetic field . Some of the field lines
of will pass through the second loop. Let be the flux
of through loop 2:
where 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,
that the magnitude of is proportional to the current .
ultimately a consequence of the linearity of Maxwell's equations.
Here, is a line element of loop 1 located at position
It follows that
the flux must also be proportional to . Thus, we can write
where is the constant of proportionality. This constant is called
the mutual inductance of the two loops.
Let us write the magnetic field in terms of a vector potential , so that
It follows from Stokes' theorem that
where is a line element of loop 2.
However, we know that
The above equation is just a special case of the more general law,
is the cross-sectional area of loop 1. Thus,
where is now the position vector of the line element
of loop 2. The above equation implies that
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,
In fact, we can drop the subscripts, and just call these quantities .
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 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 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 ,
is proportional to the current, so we can write
The constant of proportionality is called the self-inductance. Like
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.
Up: Magnetic induction