# ampere's-law

More Than You Ever Wanted to Know About Electrical Engineering: More About Inductance

We’ve been talking about three-phase circuits of the kind you might see in residential systems, generally at 120 or 208 VAC. The voltage you actually see on a utility line is much higher than this, though, as it is more efficient (and consequently cheaper) to transmit over long distances at very high voltages. So how do you go from the very high voltage circuits on a utility line to the much lower voltages in a commercial or residential electrical system?

The device to handle this problem is a transformer. To understand how it works, we’re going to need to take a closer look at inductors and a little bit of electromagnetic theory.

At this point, we know that an inductor is basically a coiled bit of wire that stores and releases energy in a magnetic field. Let’s examine how that works a bit more closely.

There are two physical laws that will be relevant for us here. The first is Ampere’s Law, which simply says that any time you have an electric current, you get a magnetic field. So if we pass a current through a coil of wire, we’ll produce a magnetic flux, φ.

The geometry of the coils actually amplifies this effect - the coils in close proximity to each other “catch” more of the overall magnetic flux. You’ll see a term called flux linkage, λ, get thrown around. This is just a measure of how much magnetic flux is caught up in the coil, and it’s directly proportional to both the number of turns of the coil, N, and the overall flux.

We’ve already said that the presence of the magnetic flux is related to the current flowing through the coil, so we should also expect that the flux linkage is proportional to current. The constant of proportionality between the two is the coil’s inductance. This is all inductance really is: a measure of the relationship between the current flowing through a coil and the magnetic flux it generates.

The other physical law of interest to us here is Faraday’s Law, which states that anytime we have a changing magnetic flux, it creates a voltage.

We can substitute in our equation linking flux linkage, inductance, and current here. Applying the chain rule, we get something that starts to look a little familiar.

In most circumstances, inductance is a constant. We don’t have to worry about it changing over time, so for our purposes, this equation reduces to the equation relating voltage, current, and inductance which we’ve used before.

“Use Ampere’s law to derive an expression for the magnetic ﬁeld B inside the cylinder in terms of r, R, I, and fundamental constants”

How about I use newton’s laws to derive an expression for the acceleration of my foot into a silly little frenchman's ass for making physics so goddamn frustrating