Inductance

To find the inductance (L) of a coil, find the magnetic flux (field times area) through each loop, add them, and divide by current: L = (S f )/I. (Some books define inductance as Nf /I, which would be OK if you use average flux.) Flux is proportional to I for an air-core coil, so L is independent of I. For an iron core coil, inductance is not really a constant, but we treat it as a constant and there is not much error. (I have not seen anything about the amount of error involved.)

For a long coil of length d and radius r, a good approximation to the inductance is
L = m _o N² p r²/d. This is based on two approximations with errors in opposite directions, so it works ok. The truth is that
f > m _o NI p r²/d at the center of the coil, and
f < m _o NI p r²/d at the end of the coil, so for the derivation, one assumes an
average f = m _o NI p r²/d, and it is reasonably close, not exact. Now to get L from this, multiply that average f by N and divide by I.

The purpose of inductance is for applying Faraday's law to a change of flux in a coil caused by a change in the current of the coil itself: self induction. The voltage induced is the sum of the voltages in the loops, or

e = S (D f /D t) or S df /dt in calculus lingo.

Replace S f with LI from our definition of inductance, and use the sign convention discussed below, and we have

e = -LdI/dt

It turns out that if you go in the direction of current through a coil, the sign of the voltage change is opposite to the sign of dI/dt, hence the minus sign. The voltage is such that it opposes changes of current, so it makes the current act as if it has inertia.

An LR circuit

The voltage equation for the circuit below, going clockwise around the loop when the switch is on, is e - IR - LdI/dt = 0. It is easy to show from this that if the switch is turned on at t = 0, I = (e /R)(1 - e^-t/t ), where t = L/R, the "time constant."

If the switch above is opened again after the current is built up, a very high voltage can be induced in the inductor because of the extremely rapid change of I. This is bad for switches, because arcing can occur. It can also be a shocking experience.

Any circuit has inductance. A simple battery-light bulb circuit is one loop, so the current will produce flux through the loop, and L = flux/current (but calculating it may be difficult).

The unit of inductance is the henry (H). Go back to the main E&M page, this should only take a few H/W .

Other main physics pages:

mechanics
vibrations and waves
quantum

Comments, questions: fredrickgram