Faraday’s experiments and magnetic flux
A magnet, a coil and a needle
We know that an electric current produces a magnetic field. Around 1830, Michael Faraday in England and Joseph Henry in America asked the reverse question: can a magnet produce a current?
Here is Faraday’s set-up. A coil of wire is joined to a galvanometer, a sensitive meter whose needle swings left or right for a current in either direction. There is no battery anywhere.
Push the north pole of a bar magnet towards the coil. The needle deflects. A current flows, with no battery in the circuit!
Now hold the magnet still, even right inside the coil. The needle goes back to zero. A magnet at rest gives no current, however strong it is.
Pull the magnet away, and the needle deflects the other way. Bring in the south pole instead, and every deflection is reversed.
Move the magnet faster, and the deflection is larger. Keep the magnet still and move the coil instead: exactly the same thing happens. Only the relative motion matters.
Faraday then replaced the magnet with a second coil carrying a steady current from a battery. Moving this coil towards or away from the first one also deflects the needle.
Finally, he kept both coils still and put a tap key in the battery circuit. Press the key: the needle kicks, then returns to zero. Release it: the needle kicks the other way.
Put a soft iron rod through both coils, and the kicks become much larger.
What is common to all these experiments? The current appears only while the magnetic field through the coil is changing. To say this exactly, we need a new quantity: magnetic flux.
Magnetic flux
Think of magnetic field lines passing through a window. The magnetic flux through the window measures how much field passes through it.
Take a flat area A in a uniform field B. Draw the area vector: its length is A, and it points along the normal to the surface. Let θ be the angle between B and this normal.
The magnetic flux is the dot product: Φ_B equals B dot A, which is B A cos θ.
When the field is along the normal, θ is 0 and the flux is a maximum, B A. Tilt the surface, and fewer field lines pass through it. When the field lies in the plane of the surface, θ is 90°, and the flux is zero.
Turn the surface further, to θ equals 180°, and the flux becomes minus B A. The field now goes through from the other side.
If the field is not uniform, cut the surface into tiny pieces dA. Add up B dot dA for every piece. Flux is a scalar.
The SI unit of flux is the weber, written Wb. One weber is one tesla metre squared.
An example. A loop of area 50 centimetre squared, that is 5 × 10⁻³ metre squared, is in a field of 0.2 tesla, at 60° to its normal. Φ_B equals 0.2 × 5 × 10⁻³ × cos 60°, which is 5 × 10⁻⁴ weber.
So there are three ways to change the flux: change B, change the area A, or change the angle θ. In each of Faraday’s experiments, one of these was changing.
And whenever the flux through the coil changed, a current flowed. In the next lesson, we find exactly how big that effect is.
Try it: Faraday’s magnet and coil
- Drag the magnet into the coil and watch the galvanometer. Then hold it still inside: the needle returns to zero.
- Pull it out: the needle swings the other way. Switch to “S faces coil” and every deflection reverses.
- Use the buttons at 10 cm/s and then at 40 cm/s. Faster motion gives a bigger kick, but for a shorter time.
- The top graph is the flux linkage NΦ_B. The emf appears only while that graph is sloping.
A strong magnet at rest
A very strong bar magnet is held at rest inside a coil that is connected to a galvanometer. What does the galvanometer show?
- A large steady deflection, because the field inside the coil is strong
- Zero, because the flux through the coil is not changing
- A deflection that slowly dies away
- A deflection whose size depends on which pole is inside
Answer
B. A current is induced only while the magnetic flux through the coil changes. A magnet at rest gives a large but constant flux, so there is no emf and no current. Only motion of the magnet (or the coil), or a change in the field, gives a deflection.
Flux through a tilted loop
A square loop of side 10 cm is placed in a uniform field of 0.5 T. The field makes an angle of 60° with the plane of the loop. What is the magnetic flux through the loop?
- 2.5 × 10⁻³ Wb
- 4.33 × 10⁻³ Wb
- 5.0 × 10⁻³ Wb
- 43.3 Wb
Answer
B. θ in Φ_B = BA cos θ is measured from the normal, not from the plane. The field is at 60° to the plane, so it is at 30° to the normal. A = (0.10 m)² = 10⁻² m². Φ_B = 0.5 × 10⁻² × cos 30° = 0.5 × 10⁻² × 0.866 = 4.33 × 10⁻³ Wb. Using 60° gives 2.5 × 10⁻³ Wb; ignoring the angle gives 5.0 × 10⁻³ Wb; using A = 100 cm² without converting gives 43.3.
Summary
- Magnetic flux (uniform B, flat area):
Φ_B = B · A = BA cos θ - General case:
Φ_B = Σ B · dA - Unit:
1 weber (Wb) = 1 T m²
- Faraday and Henry found that a current is induced in a coil whenever the magnetic field through it changes: a moving magnet, a moving current-carrying coil, or a current switched on or off nearby.
- A magnet or a steady current at rest induces nothing. Faster changes give bigger deflections; only relative motion matters.
- Magnetic flux Φ_B = BA cos θ, where θ is the angle between B and the normal to the surface. It is a scalar.
- Flux can be changed in three ways: change B, change the area A, or change the angle θ.