Faraday’s Law
9.1 Faraday’s Law
Learning Objectives
By the end of this section, you will be able to:
- Determine the magnetic flux through a surface, knowing the strength of the magnetic field, the surface area, and the angle between the normal to the surface and the magnetic field
- Use Faraday’s law to determine the magnitude of induced emf in a closed loop due to changing magnetic flux through the loop
Faraday’s Law
The emf 𝜀 induced is the negative change in the magnetic flux Φm per unit time. Any change in the magnetic field or change in orientation of the area of the coil with respect to the magnetic field induces a voltage (emf).
Faraday’s Law of Induction
Let’s start with defining what we mean by the word “induction.” If you look it up in the dictionary, you’d find something like “the process or action of bringing about or giving rise to something.” So when we talk specifically about Faraday’s law of induction, it describes the action of bringing about or giving rise to an emf.
Practice!
| Practice 9.1.1 |
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Consider the situation in the figure. A wire is connected to the input and ground terminals of a voltmeter and formed into a circular loop with two turns. A bar magnet is held at the center of the loop. In order to change the magnetic flux through the loop, what could you do? |
| Practice 9.1.2 |
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| Consider the situation in the figure. A wire is connected to the input and ground terminals of a voltmeter and formed into a circular loop with two turns. A bar magnet is held at the center of the loop. In order to change the magnetic flux through the loop, what else could you do? |
Faraday’s Law says it is only the change in flux through the loop that matters. A huge B-field (lots of flux) does NOT make the emf, it’s the change in B with time that does the trick. Since Φ = BAcosθ, you can change the flux in many ways: you could change B, or the area, or the angle between B and the loop. These changes in the flux would all result in an induced emf in the coil.
There is a minus sign in Faraday’s law, which we will discuss shortly. But I want to put aside for the moment. Suffice it to say, that minus sign tells you about the direction of the induced emf and the direction that the induced current will flow in the loop. This is called Lenz’s law, and I will discuss it in detail in the next part of the module. What we can do for now is find the magnitude of the induced emf and not worry about its direction yet.
Practice!
Two circular loops of wire with small bulbs in them are sitting beside two long straight current-carrying wires. The loops with the bulbs have no battery or power supply, they are simply wires with bulbs. The long straight wires are connected to batteries not shown, but that is where the currents I1 and I2 come from. Recall from the last module that currents in a long straight wire produce magnetic fields that circulate around them with a magnitude of 
. The wire loops, bulbs, and long straight wires are identical for the two cases. The loops are the same distance from the straight wires. But situation A is isolated from situation B, they have no effect on each other. We are going to look at different scenarios and predict what will happen to the bulb.
In each of the following questions, determine if the bulbs will light up.
| Practice 9.1.3 |
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There is a constant current in wire A of 6 A (amperes) and a constant current in wire B of 15 A. |
| Practice 9.1.4 |
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| Both wires start with the same initial 3-A current, but the current in wire B increases to 6 A in a 0.3 second interval while the current in A remains constant. |
| Practice 9.1.5 |
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| The current in wire A goes from 2 A to 10 A in a 0.5 second interval, while at the same time, the current in B goes from 12 A to 16 A. |
| Practice 9.1.6 |
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| The current in wire A decreases from 10 A to 4 A in a 0.2 second interval, while the current in wire B increases from 9 A to 18 A in 0.3 seconds. |
| Practice 9.1.7 |
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| The current in wires A and B both double in a 0.2 second interval, but the current in wire A starts at twice the initial value of B. |
| Pause & Predict 9.1.1 |
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| What is the change in magnetic flux through the loop during this time interval? |
| Pause & Predict 9.1.2 |
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| What is the induced current? |
Practice!
| Practice 9.1.8 |
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| A 144-Ω light bulb is connected to a conducting wire that is wrapped into the shape of a square with side length of 83.0 cm. This square loop is rotated within a uniform magnetic field of 454 mT. What is the change in magnetic flux through the loop when it rotates from a position where its area vector makes an angle of 30° with the field to a position where the area vector is parallel to the field? |
| Practice 9.1.9 |
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| A 144-Ω light bulb is connected to a conducting wire that is wrapped into the shape of a square with side length of 83.0 cm. This square loop is rotated within a uniform magnetic field of 454 mT. The loop rotates from a position where its area vector makes an angle of 30° with the field to a position where the area vector is parallel to the field in 56.3 ms. What is the induced current through the light bulb? |
| Practice 9.1.10 |
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| A 144-Ω light bulb is connected to a conducting wire that is wrapped into the shape of a square with side length of 83.0 cm. This square loop is rotated with a frequency of 60 Hz within a uniform magnetic field of 454 mT. This means the loop makes half a revolution in 8.33 ms. What is the induced current in the light bulb when the loop rotates from a position where its area vector is opposite the magnetic field to a position where its area vector is parallel to the magnetic field? |
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