Lesson 10 of 10 · 15 min

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Must-know facts

17 facts

1An emf is induced only while the magnetic flux through a circuit is changing.
2ΦB = B·A = BA cos θ, θ between B and the normal; unit weber, 1 Wb = 1 T m².
3Faraday's law: ε = −N dΦB/dt for a coil of N turns.
4Flux changes if B, the area or the angle between B and the normal changes.
5Lenz's law: the induced current opposes the change in flux that causes it; this follows from energy conservation.
6An N pole approaching a coil makes the near face an N pole (repulsion); receding, an S pole (attraction).
7Motional emf across a rod moving at right angles to B: ε = Blv.
8Rod of length R rotating about one end with angular speed ω: ε = ½BωR².
9A time-varying magnetic field produces an electric field.
10Flux linkage NΦ = LI (self) or N₁Φ₁ = MI₂ (mutual); unit henry, dimensions [M L² T⁻² A⁻²].
11Long coaxial solenoids: M = μ₀n₁n₂πr₁²l; M₁₂ = M₂₁.
12Long solenoid: L = μ₀n²Al, or μrμ₀n²Al with a core.
13Self-induced emf ε = −L dI/dt opposes both rise and fall of current.
14Energy stored in an inductor: ½LI²; magnetic energy density B²/2μ₀.
15AC generator: ε = NBAω sin ωt, peak ε₀ = NBAω.
16Generator emf is maximum when the coil's plane is parallel to B and zero when it faces B.
17Supply frequency: 50 Hz in India, 60 Hz in the USA.

Common traps

Where marks are lost

Writing the flux as BA sin θ with θ measured from the normal.

Φ = BA cos θ when θ is the angle between B and the area vector (the normal). If the angle is given with the plane, use sin of that angle.

Expecting a current in a loop held still between very strong magnets.

Only a changing flux induces an emf. A steady field of any strength gives nothing.

Forgetting the number of turns in ε = −N dΦ/dt.

Each turn adds the same emf, so multiply the per-turn flux change by N.

Using BωR² for a rotating rod.

Points on the rod move at ωr, not ωR; averaging over the length gives ε = ½BωR².

Adding the emfs of all the spokes of a rotating wheel.

The spokes are in parallel between axle and rim, so the emf is that of one spoke.

Thinking the back emf only opposes a rising current.

ε = −L dI/dt opposes any change: it tries to keep a falling current going too.

Placing the generator's peak emf where the flux is largest.

The emf is largest where the flux is zero and changing fastest, with the coil's plane along B.

Taking the flux change as zero when a coil is turned through 180°.

The flux goes from +BA to −BA, so the change is 2BA.

Believing a changing electric flux through a loop drives a current in it.

Faraday's law concerns magnetic flux; a loop moving in a steady electric field has no induced current.

Formulas

10 to know

Magnetic flux

ΦB = B·A = BA cos θ

θ between B and the normal; unit Wb = T m².

Faraday's law

ε = −N dΦB/dt

Minus sign is Lenz's law.

Motional emf

ε = Blv

Rod, velocity and field mutually perpendicular.

Rotating rod

ε = ½BωR²

Between pivot and free end; field along the axis.

Mutual inductance, coaxial solenoids

M = μ₀n₁n₂πr₁²l

r₁ is the inner radius; multiply by μr for a core.

Mutually induced emf

ε₁ = −M dI₂/dt

M₁₂ = M₂₁.

Self-inductance of a solenoid

L = μrμ₀n²Al

μr = 1 for air.

Self-induced emf

ε = −L dI/dt

Back emf.

Energy in an inductor

U = ½LI², u = B²/2μ₀

u is energy per unit volume.

AC generator emf

ε = NBAω sin ωt, ε₀ = NBAω

ω = 2πν; θ = 0 at t = 0.

Key terms

16 terms

Electromagnetic induction: Producing an emf, and a current in a closed circuit, by changing the magnetic flux through it.
Magnetic flux (ΦB): B·A summed over a surface; a scalar measured in webers.
Weber (Wb): SI unit of magnetic flux, 1 T m².
Faraday's law: Induced emf equals minus the rate of change of flux linkage.
Lenz's law: The induced current flows so as to oppose the change in flux that produced it.
Motional emf: The emf Blv across a conductor moving through a magnetic field.
Flux linkage: NΦB for a closely wound coil of N turns.
Inductance: Flux linkage per unit current; depends on geometry and core material.
Henry (H): SI unit of inductance, 1 Wb/A.
Mutual inductance (M): Flux linkage in one coil per unit current in another; the same either way round.
Self-inductance (L): A coil's own flux linkage per unit current in it; electrical inertia.
Back emf: The self-induced emf −L dI/dt that opposes changes in current.
Magnetic energy density: Energy stored per unit volume in a magnetic field, B²/2μ₀.
AC generator: A machine that turns a coil (or magnets) to make an alternating emf NBAω sin ωt.
Armature: The rotating coil of a generator.
Slip rings and brushes: Sliding contacts that carry the generator coil's current to the outside circuit.

Test yourself: 10 questionsExam-style questions on Electromagnetic Induction, with full solutions.Start