So far, we talked about how you can create a curly electric field (and thus an induced voltage/induced current) from a changing magnetic flux. We have gone through examples of what happens when the area changes and in your last project you worked through what happens when the coil is rotating. You also had a demo video that showed what happened when a strong magnet was brought towards or away from the coil, which is one way that the strength of the magnetic field can change. If instead, you have a magnetic field that is produced by a current, another way that the magnetic field can change is if the current (that is producing the magnetic field) changes. This is actually a fairly common situation for two reasons: 1) we are often turning on and off electrical devices and 2) the current/voltage coming from the wall outlets is an alternating current (meaning it is constantly changing from positive to negative). These notes are going to focus on the first of these, that is, how an induced current develops from a magnetic field that is changing with time that is not the result of the physical motion of a magnet. (The next page of notes will discuss the second case.)

When you initially connect a resistor to a battery, the current in the circuit is initially zero and then increases to its steady state value. If we made a graph of the magnitude of the current in the wires around the circuit, it would look something like one shown to right. In this case we would mathematically model the current as $I=I_0(1-e^{-t/\tau})$, where $I_0$ is the steady state current and $\tau$ is a constant that tells you how fast the current reaches the steady state. If you only have a resistor in the circuit, it takes nano-seconds to micro-seconds to reach the steady state current.

While this is a realistic representation of the current in the circuit, mathematically, it can be a little complicated. Instead, we will simplify the model and approximate the increase of current in the circuit as shown in the graph to right. We will say that there is a period of time where the current is increasing linearly, and after that time we have a constant steady state current.

When the current is at its steady state value, there would not be an induced current in a nearby coil because the magnetic field would not be changing (constant current would mean the magnetic field would also be constant over time). However, when the current is increasing, you would have an increasing magnetic field around the wires. If you were to put a loop of wire near this increasing magnetic field, you would see an induced current in the loop, even though the wire is not physically moving!

When you have a typical circuit (like a battery connected to a light bulb), the change in the magnetic field is small enough that the induced current in any nearby loop is probably negligible. However, as circuit components become smaller and smaller and are placed closer and closer together (with roughly the same amount of current), the induced currents (because of changing magnetic fields) can become an important consideration in the design of electronics.

As an example, let's consider a set of concentric coils, where the larger outside coil is initially connected to a battery so that it's current increases from 0 A to 1 A in 1 ns. What would be the induced potential in the smaller inner coil?

We know from Faraday's law that the induced voltage in the small loop should be equal to the change in magnetic flux through the small coil. Since the small coil is at the center of the large coil, the magnetic flux through the small coil would be due to the magnetic field from the large coil.

This means we want to start by writing out what would be the magnetic field at the center of the larger loop (and therefore the magnetic field going through the small coil). We found in an earlier example that the magnetic field through a coil (with a counter-clockwise current) would be:

$$\vec{B}_{lg}=\frac{\mu_0N_{lg}I}{2 R_{lg}}\hat{z}$$ where $N_{lg}$ is the number of turns of wire around the large coil, $I$ is the current through the large coil and $R_{lg}$ is the radius of the large coil. If we assume that the current increases linearly from 0 A to 1 A in 1 ns, this means that we can write the current as: $$I = \frac{\Delta I}{\Delta t} t$$ $$I = \frac{1}{1\cdot10^{-9}}t=1\cdot 10^9 t$$ We can plug