Induced EMF for AP Physics B & C
Practice problems for these concepts can be found at:
A changing magnetic field produces a current. We call this occurrence electromagnetic induction.
So let's say that you have a loop of wire in a magnetic field. Under normal conditions, no current flows in your wire loop. However, if you change the magnitude of the magnetic field, a current will begin to flow.
We've said in the past that current flows in a circuit (and a wire loop qualifies as a circuit, albeit a simple one) when there is a potential difference between the two ends of the circuit. Usually, we need a battery to create this potential difference. But we don't have a battery hooked up to our loop of wire. Instead, the changing magnetic field is doing the same thing as a battery would. So rather than talking about the voltage of the battery in this circuit, we talk about the "voltage" created by the changing magnetic field. The technical term for this "voltage" is induced EMF.
For a loop of wire to "feel" the changing magnetic field, some of the field lines need to pass through it. The amount of magnetic field that passes through the loop is called the magnetic flux. This concept is pretty similar to electric flux.
The units of flux are called webers; 1 weber = 1 T·m2. The equation for magnetic flux is
In this equation, ΦB is the magnetic flux, B is the magnitude of the magnetic field, A is the area of the region that is penetrated by the magnetic field.
Let's take a circular loop of wire, lay it down on the page, and create a magnetic field that points to the right, as shown in Figure 22.9.
No field lines go through the loop. Rather, they all hit the edge of the loop, but none of them actually passes through the center of the loop. So we know that our flux should equal zero.
Okay, this time we will orient the field lines so that they pass through the middle of the loop. We'll also specify the loop's radius = 0.2 m, and that the magnetic field is that of the Earth, B = 5 × 10–5 T. This situation is shown in Figure 22.10.
Now all of the area of the loop is penetrated by the magnetic field, so A in the flux formula is just the area of the circle, πr2.
The flux here is
- ΦB = (5 × 10–5)(π) (0.22) = 6.2 × 10–6 T·m2.
Sometimes you'll see the flux equation written as BAcosθ. The additional cosine term is only relevant when a magnetic field penetrates a wire loop at some angle that's not 90°. The angle θ is measured between the magnetic field and the "normal" to the loop of wire … if you didn't get that last statement, don't worry about it. Rather, know that the cosine term goes to 1 when the magnetic field penetrates directly into the loop, and the cosine term goes to zero when the magnetic field can't penetrate the loop at all.
Because a loop will only "feel" a changing magnetic field if some of the field lines pass through the loop, we can more accurately say the following: A changing magnetic flux creates an induced EMF.
Faraday's law tells us exactly how much EMF is induced by a changing magnetic flux.
ε is the induced EMF, N is the number of loops you have (in all of our examples, we've only had one loop), and Δt is the time during which your magnetic flux, ΦB, is changing.
Up until now, we've just said that a changing magnetic flux creates a current. We haven't yet told you, though, in which direction that current flows. To do this, we'll turn to Lenz's Law.
When a current flows through a loop, that current creates a magnetic field. So what Lenz said is that the current that is induced will flow in such a way that the magnetic field it creates points opposite to the direction in which the already existing magnetic flux is changing.
Sound confusing?4 It'll help if we draw some good illustrations. So here is Lenz's Law in pictures.
We'll start with a loop of wire that is next to a region containing a magnetic field (Figure 22.11a). Initially, the magnetic flux through the loop is zero.
Now, we will move the wire into the magnetic field. When we move the loop toward the right, the magnetic flux will increase as more and more field lines begin to pass through the loop. The magnetic flux is increasing out of the page—at first, there was no flux out of the page, but now there is some flux out of the page. Lenz's Law says that the induced current will create a magnetic field that opposes this increase in flux. So the induced current will create a magnetic field into the page. By the right-hand rule, the current will flow clockwise. This situation is shown in Figure 22.11b.
After a while, the loop will be entirely in the region containing the magnetic field. Once it enters this region, there will no longer be a changing flux, because no matter where it is within the region, the same number of field lines will always be passing through the loop. Without a changing flux, there will be no induced EMF, so the current will stop. This is shown in Figure 22.11c.
To solve a problem that involves Lenz's Law, use this method:
- Point your right thumb in the initial direction of the magnetic field.
- Ask yourself, "Is the flux increasing or decreasing?"
- If the flux is decreasing, then just curl your fingers (with your thumb still pointed in the direction of the magnetic field). Your fingers show the direction of the induced current.
- If flux is increasing in the direction you're pointing, then flux is decreasing in the other direction. So, point your thumb in the opposite direction of the magnetic field, and curl your fingers. Your fingers show the direction of the induced current.
Induced EMF in a Rectangular Wire
Consider the example in Figures 22.11a–c with the circular wire being pulled through the uniform magnetic field. It can be shown that if instead we pull a rectangular wire into or out of a uniform field B at constant speed v, then the induced EMF in the wire is found by
Here, L represents the length of the side of the rectangle that is NOT entering or exiting the field, as shown below in Figure 22.12.
Some Words of Caution
We say this from personal experience. First, when using a right-hand rule, use big, easy-to-see gestures. A right-hand rule is like a form of advertisement: it is a way that your hand tells your brain what the answer to a problem is. You want that advertisement to be like a billboard—big, legible, and impossible to misread. Tiny gestures will only lead to mistakes. Second, when using a right-hand rule, always use your right hand. Never use your left hand! This will cost you points!
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