Faraday’s Law of Electromagnetic Induction: MDCAT Physics notes
Faraday’s Law of Electromagnetic Induction for MDCAT: induced emf = −NΔΦ/Δt, motional emf BLv, AC generator emf, Fleming's right-hand rule and units.
Electromagnetic induction
An emf is induced in a circuit whenever the magnetic flux linking it changes. A steady flux, however large, induces nothing: a stationary magnet held near a coil gives no galvanometer deflection. The flux can be changed by moving a magnet or coil, by changing the current in a nearby coil, or by rotating a coil in a field.
Faraday's law
The induced emf equals the rate of change of flux linkage:
$$\varepsilon = -N\frac{\Delta\Phi}{\Delta t}$$
- It depends on the rate of change, not on the flux itself or merely on whether it increases or decreases.
- The minus sign expresses Lenz's law (the emf opposes the change).
- $\Delta\Phi/\Delta t$ has the unit Wb s$^{-1}$ = volt = N m A$^{-1}$ s$^{-1}$.
Example: a 200-turn coil whose flux falls from 0.05 Wb to 0.01 Wb in 0.2 s has $\varepsilon = 200\times0.04/0.2 = 40$ V.
Motional emf
A rod of length $L$ moving with velocity $v$ perpendicular to a field $B$ sweeps out area, and the emf across it is
$$\varepsilon = BLv \quad (\text{generally } BLv\sin\theta)$$
It depends on the field, the length, the speed and the orientation. Example: $B = 0.4$ T, $L = 0.5$ m, $v = 10$ m s$^{-1}$ gives 2 V. Convert centimetres to metres first.
Fleming's right-hand rule (generators): thumb = motion, first finger = field, second (middle) finger = induced current.
The AC generator
A generator converts mechanical energy into electrical energy. A coil of $N$ turns and area $A$ rotates with angular speed $\omega$ between two magnetic poles (N and S). The flux through it varies as $\Phi = BA\cos\omega t$, giving
$$\varepsilon = NAB\omega\sin\omega t = \varepsilon_0\sin\omega t$$
| Plane of coil | Flux | Rate of change of flux | emf |
|---|---|---|---|
| Perpendicular to $B$ | Maximum | Zero | Zero |
| Parallel to $B$ | Zero | Maximum | Maximum ($\sin\omega t = 1$) |
The emf reverses every half turn, so the output is alternating and continuously changing. A generator is essentially a motor run in reverse.
Rotating a hoop
To induce an emf the rotation must change the angle between the field and the plane of the loop. Rotating a loop about an axis parallel to the field, or sliding it through a uniform field without rotating, leaves the flux unchanged and induces nothing.
Self-inductance and back emf
A changing current in a coil induces an emf in the same coil that opposes the change; it is called the back emf. The self-inductance of a coil depends on its number of turns, area, length and core material, not on the current or the resistance of the wire.
Key formulas
- $\varepsilon = -N\Delta\Phi/\Delta t$
- $\varepsilon = BLv\sin\theta$
- $\varepsilon = NAB\omega\sin\omega t$, $\varepsilon_0 = NAB\omega$
Common MDCAT traps
- The emf depends on the rate of change of flux, not the amount of flux.
- Emf is maximum when the plane of the coil is parallel to $B$, even though flux is then zero.
- Middle finger in the right-hand rule is induced current, not force.
- No emf from a stationary magnet, however strong.
- Generators and transformers both work on Faraday's law.
Quick revision
- Changing flux is the basic requirement for induced emf.
- Generator: mechanical to electrical energy.
- Rate of change of flux is measured in volts.
- AC generators cannot produce DC directly.
- Induced current flows only in a closed circuit; emf appears even in an open one.