Magnetism: Of Special Interest to Physics C Students
Practice problems for these concepts can be found at:
Important Consequences of Biot–Savart and Ampére's Law
So far we've only discussed two possible ways to create a magnetic field—use a bar magnet, or a long, straight, current-carrying wire. And of these, we only have an equation to find the magnitude of the field produced by the wire.
The Biot–Savart law provides a way, albeit a complicated way, to find the magnetic field produced by pretty much any type of current. It's not worth worrying about using the law because it's got a horrendously complicated integral with a cross product included. Just know the conceptual consequence: A little element of wire carrying a current produces a magnetic field that (a) wraps around the current element via the right-hand rule, and (b) decreases in magnitude as 1/r2, r being the distance from the current element.
So why does the magnetic field caused by a long, straight, current-carrying wire drop off as 1/r rather than 1/r2? Because the 1/r2 drop-off is for the magnetic field produced just by a teeny little bit of current-carrying wire (in calculus terminology, by a differential element of current). When we include the contributions of every teeny bit of a very long wire, the net field drops off as 1/r.
Ampére's law gives an alternative method for finding the magnetic field caused by a current. Although Ampére's law is valid everywhere that current is continuous, it is only useful in a few specialized situations where symmetry is high. There are two important results of Ampére's law:
- The magnetic field produced by a very long, straight current is
- A solenoid is set of wound wire loops. A current-carrying solenoid produces a magnetic field. Ampére's law can show that the magnetic field due to a solenoid is shaped like that of a bar magnet; and the magnitude of the magnetic field inside the solenoid is approximately uniform, Bsolenoid = μ0 nI. (Here I is the current in the solenoid, and n is the number of coils per meter in the solenoid.)
- The magnetic field produced by a wire-wrapped torus (a "donut" with wire wrapped around it [see Figure 22.13]) is zero everywhere outside the torus, but nonzero within the torus. The direction of the field inside the torus is around the donut.
outside the wire; inside the wire, the field increases linearly from zero at the wire's center.
Inductors in Circuits
An inductor makes use of induced EMF to resist changes in current in a circuit. If part of a circuit is coiled, then the magnetic field produced by the coils induces a "back EMF" in the rest of the circuit … that EMF depends on how fast the current is changing, by Faraday's law. An inductor in a circuit is drawn as a little coil, as shown in Figure 22.14.
The voltage drop across an inductor is
where L is called the inductance of the inductor. Inductance is measured in units of henrys.
What does this equation mean? If the current is changing rapidly, as when a circuit is first turned on or off, the voltage drop across the inductor is large; if the current is barely changing, as when a circuit has been on for a long time, the inductor's voltage drop is small.
We can think of an inductor as storing energy in the magnetic field it creates. When current begins to flow through the inductor, it stores up as much energy as it can. After a while, it has stored all the energy it can, so the current just goes through the inductor without trouble. The energy stored in an inductor is found by this equation.
For the AP Physics C exam, you need to understand circuits with inductors and resistors, as well as circuits with inductors and capacitors.
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