## Homework Helpers: Physics

## 7 Magnetism

### Lesson 7–4: Electromagnetic Induction

As I mentioned in __Lesson 7–2,__ Hans Christian Oersted showed that electricity could generate a magnetic field. Scientists then began to wonder if the reciprocal were also true. Could magnetic fields generate electricity? Through trial and error, two scientists eventually found that they could induce a current in a wire loop by moving a magnet back and forth through the center of it. Further experimentation showed that moving or turning a wire loop in a magnetic field will also induce a current. The two scientists were Michael Faraday (1791–1867), an Englishman, and an American named Joseph Henry (1797–1878). They share the credit for independently discovering electromagnetic induction, the process of generating electricity with a changing magnetic field. Very few discoveries, if any, have had a greater impact on our way of life.

Faraday, who is also responsible for the mental model of fields of force, concluded that when the number of magnetic field lines through the wire loop changes, a current is induced. Further, the magnitude of the induced current is proportional to the rate of this change. In other words, you can induce a stronger current in a wire loop by moving a bar magnet quickly through it. Because a stronger bar magnet would have more field lines per unit of area, you get more induced current with a stronger magnet. You could also increase the number of loops or turns in the wire coil. In summary, you can say that the current is induced by any change in **magnetic flux**.

**Magnetic Flux**

The product of the magnetic field and the area through which the magnetic field lines pass.

ϕ = BA cos*θ*

Where ϕ = the magnetic flux, B = the magnetic field, A = the area the field passes through, and *θ* = the angle between the field lines and the surface of the area they pass through.

Magnetic flux is measured in weber (Wb), where 1 Wb = 1 T × m^{2}.

Remember: Newton”s second law tells us that the acceleration of an object is inversely proportional to its mass. Based on your understanding of Newton”s second law, you will probably not be surprised to learn that it is often easier to turn a wire in a magnetic field than it is to move a bar magnet into and out of a wire coil. For this reason, many of today”s generators employ this technique in order to induce current. Let”s see how rotating a wire loop in an unchanging magnetic field can result in a change in magnetic flux.

**Example 1**

A square-shaped wire loop that is 3.0 cm long and 2.0 cm wide rests in a uniform 1.45 T magnetic field. The magnetic field is perpendicular to the surface of the area of the wire loop. Find the magnetic flux.

**Calculate:** A = L × W = (0.030 m)(0.020 m) = 6.0 × 10^{-4} m^{2}

Note: If the wire loop is perpendicular to the magnetic field, then the angle between the field and the surface of the area is zero.

**Find:** ϕ

**Solution:** ϕ = BAcos*θ* = (1.45 T)(6.0 × 10^{-4} m^{2})(cos 0°)

= 8.7 × 10^{-4} Wb

The value of cosine 0° = 1, so the magnetic flux through the wire loop is maximized when the field lines are perpendicular to the wire loop, meaning the angle between the field lines and the surface area is zero. What would happen if we took the same magnetic field and wire loop from__Example 1__ and rotated the wire loop so that the angle between the field lines and the surface of the area was 45°? Let”s try that as our second example.

**Example 2**

A square-shaped wire loop that is 3.0 cm long and 2.0 cm wide rests in a uniform 1.45 T magnetic field. The angle between the field lines and the surface of the area of the wire loop is 45°. Find the magnetic flux.

**Find:** ϕ

**Solution:** ϕ = BAcos*θ* = (1.45 T)(6.0 × 10^{-4} m^{2})(cos 45°)

= 6.2 × 10^{-4} Wb

As you can see in __Example 2__, changing the angle of the wire loop decreased the magnetic flux. This makes sense because the number of field lines actually going through the wire loop would have decreased. We can calculate the change in magnetic flux between __examples 1__ and __2__:

Δϕ = ϕ_{f} – ϕ_{i} = (6.2 × 10^{-4} Wb) – (8.7 × 10^{-4} Wb) = – 2.5 × 10^{-4} Wb.

**emf**

I have avoided introducing **emf** until now, but we need to go over it in order to introduce Faraday”s law of magnetic induction.

The letters *emf* actually stand for electromotive force. However, it turns out that the so-called “electromotive force” is not actually a force, so the initials emf are now more commonly used. Emf can be thought of as the voltage that is induced by magnetic induction. Emf is measured in volts (V) and is represented by the symbol *ε*.

**Faraday”s Law of Magnetic Induction**

The instantaneous emf resulting from magnetic induction equals the rate of change of flux.

where N = number of wire loops in a wire coil.

**Example 3**

Use Faraday”s law of induction to determine the emf that would be generated in our previous examples, as we turned our wire loop 45° in 0.025 s.

Notice that we divided a negative by a negative to get an answer with a positive sign. You might wonder about the significance of the sign in our answer. Before I explain it, let”s go over the implications of what we just learned. You learned that charges in motion produce magnetic fields. You also learned that changing magnetic fields can generate emf, which will, in turn, generate electricity. This new electricity will generate a magnetic field, which, as it springs into being, can produce an emf. You might be tempted to think that we could violate the law of conservation of energy in this way, but that is not the case. Lenz”s law will stop us!

**Lenz”s Law**

An induced emf gives rise to a current whose magnetic field opposes the change in magnetic flux that produced it.

What Lenz”s law actually means is that we can”t get something for nothing. If we move a wire loop in a magnetic field, we will generate current. However, the current will give rise to a magnetic field that will oppose the change that produced it. Lenz”s law is the reason why we still need to put work into turning a wire in a magnetic field to generate electricity. People have come up with creative ways of turning generators, including making use of wind and falling water.

Lesson 7–4 Review

__1.__ _______________ is the product of the magnetic field and the area through which the magnetic field lines pass.

__2.__ A magnetic flux of 1.72 Wb results from a wire loop with an area of 2.00 m orientated 35.0° from the field lines of a magnetic field going through it. Find the strength of the field.

__3.__ How many wire loops would be required in a wire coil to generate an emf of 0.078 V when the change in magnetic flux through the coil is –7.5 × 10^{-4} Wb every 0.50 s?