Springs, Capacitors and Devices 3
There are three basic circuit elements (ignoring voltage sources, current sources and the various semi-conductor devices like diodes): resistors, capacitors and inductors.
Resistors, as we know, satisfy V = IR; capacitors satisfy Q = CV (V = 1/C Q); inductors satisfy -V = L dI/dt.
The first thing to note is that they each deal with a different derivative of charge: capacitors deal with charge, resistors with the rate of change of charge, and inductors with the rate of change of rate of change of charge. So if we made a series circuit, we'd expect a second order linear differential equation to represent it. That comes with all sorts of behaviour that should be familiar: periodic oscillations, damping and resonance.
The second thing, of course, is what the heck is an inductor?
The first thing to talk about is electromagnetic induction: move a wire in a magnetic field and the electrons experience a force so long as the direction of movement is not parallel to the magnetic field. The exact direction of the force is given by the cross product F = qV X B. This is the magnetic force, or Lorentz force..
The force on the electrons creates a potential difference in the wire and causes a current to flow: now we could do some careful analysis and work out what will happen next; the current will cause a magnetic force on the wire with magnitude F = BIL, and we could use some right hand rule mess to figure out the direction. Or: Lenz's law. Lenz's law is the principle that changes in a circuit act to oppose their cause, and it prevents us from building perpetual motion machines in the following way: take a wire in a magnetic field and move it. Some current has been caused to flow. Now if Lenz's law didn't hold, then maybe that current would cause a force to accelerate the wire. That would be bad, because so long as the magnetic field holds up, we've got perpetual motion. In reality, of course, the current goes the other way and slows the wire down, decreasing the current.
This principle is used in dynamos: take a conducting disc, place it between the North and South poles of a horseshoe magnet and attach brushless contacts. Then, when the disc spins, it generates an emf due to the Lorentz force. This emf then can drive a current in whatever the disc is attached to; however, you will also get eddy currents slowing down the disc by opposing the permanent magnetic field.
But the most important use of inductance is self-inductance. Take a coil of wire and pass a current through it - from Lenz's law we know that an emf will be generated and a reverse current will flow. The important thing to do with the "back-emf" is the rate of change of current (a steady current doesn't induce an emf), so we therefore define inductance by the equation:
emf = -L dI/dt , where L is inductance
L is also defined in the following way: when a current passes through an inductor of inductance L, a flux F is formed in the inductor. Flux = Flux density * area, or F = BA for magnetic flux.
We therefore also define L as: LI = NF , where N is the number of turns in an inductor and F is the flux in each one. Alternatively (and equivalently):
L = d(NF)/d(I)
Since emf = -d(NF)/(dt) = -d(NF)/dI) * (dI/dt) = -L dI/dt
exactly as before, hence our two definitions are equivalent.
I'll leave a detailed discussion of inductors until next post, because we don't want this to be as long as the one on capacitors (!)
For now: electromagnetic induction is given by emf = -d(NF)/dt , L = d(NF)/dI , inductors are generally coils of wire (straight solenoids or tori) but like the parallel plate capacitor this is an archetypal case, not actually the general one. In the idealised model of circuits, you only have four things: a voltage source, and resistors, capacitors and inductors. This setup is sometimes known as an LRC circuit and, as we shall hopefully see next time, it behaves similarly to a spring mass system.
