Electromagnetic Induction

How a changing magnetic flux induces an EMF — Faraday's law and Lenz's law in action.

Spec Points Covered
  • I can state Faraday's law and use it to calculate induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). from changing flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns)..
  • I can state Lenz's law and explain the minus sign in Faraday's law using energyThe capacity to do work. Measured in joules (J). conservation.
  • I can derive and apply $\varepsilon = BLv$ for a straight conductor moving through a field.
  • I can derive and apply $\varepsilon = BAN\omega \sin(\omega t)$ for a coil rotating in a uniform field.
  • I can sketch and interpret graphs of flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns). and EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). against time for a rotating coil.
  • I can explain eddy currents, their energyThe capacity to do work. Measured in joules (J). dissipation, and how laminations reduce them.
Notes
01 A changing flux linkage induces an EMF — Faraday's law Faraday's law 3.7.5.4 → 02 The induced EMF opposes the change that produces it — Lenz's law Lenz's law 3.7.5.4 → 03 Four ways to induce an EMF — all change the flux linkage 3.7.5.4 → 04 EMF in a straight conductor: ε = BLv $\varepsilon = BLv$ 3.7.5.4 → 05 A rotating coil produces a sinusoidal EMF: ε = BANω sin(ωt) $N\Phi = BAN\cos(\omega t)$ 3.7.5.4 → 06 Flux linkage is cosine, EMF is sine — and they're a quarter cycle apart 3.7.5.4 → 07 Eddy currents dissipate energy as heat in bulk conductors Eddy currents 3.7.5.4 → 08 A search coil measures magnetic flux density using Faraday's law Search coil 3.7.5.4 →
Σ Key Equations Full Reference →
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Faraday's law
$$\varepsilon = -N\frac{\Delta\Phi}{\Delta t}$$
  • Where:
    • $ε$ = induced EMF (V)
    • $N$ = number of turns
    • $ΔΦ$ = change in magnetic flux (Wb)
    • $Δt$ = time interval (s)
The minus sign is Lenz's law. For magnitude calculations, use the absolute value.
EMF in a straight moving conductor
$$\varepsilon = BLv$$
  • Where:
    • $ε$ = induced EMF (V)
    • $B$ = magnetic flux density (T)
    • $L$ = length of conductor in field (m)
    • $v$ = velocity perpendicular to B (m s⁻¹)
Derived from Faraday's law by considering the area swept out: ΔΦ/Δt = B × L × v.
EMF in a rotating coil
$$\varepsilon = BAN\omega\sin(\omega t)$$
  • Where:
    • $ε$ = instantaneous EMF (V)
    • $B$ = magnetic flux density (T)
    • $A$ = area of coil (m²)
    • $N$ = number of turns
    • $ω$ = angular velocity (rad s⁻¹)
    • $t$ = time (s)
Obtained by differentiating the flux linkage NΦ = BAN cos(ωt) with respect to time.
Peak EMF (rotating coil)
$$\varepsilon_0 = BAN\omega$$
  • Where:
    • $ε₀$ = peak (maximum) EMF (V)
    • $B$ = magnetic flux density (T)
    • $A$ = area of coil (m²)
    • $N$ = number of turns
    • $ω$ = angular velocity (rad s⁻¹)
Maximum value of ε = BANω sin(ωt) when sin(ωt) = 1. The coil is edge-on to the field at this instant.
Flux linkage (rotating coil)
$$N\Phi = BAN\cos(\omega t)$$
  • Where:
    • $NΦ$ = flux linkage (Wb turns)
    • $B$ = magnetic flux density (T)
    • $A$ = area of coil (m²)
    • $N$ = number of turns
    • $ω$ = angular velocity (rad s⁻¹)
    • $t$ = time (s)
Cosine because at t = 0 the coil is face-on to the field (maximum flux linkage). Quarter cycle ahead of the EMF.
Q Retrieval Practice All 12 Questions →
Q1. State Faraday's lawThe magnitude of the induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). is proportional to the rate of change of magnetic flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns).. of electromagnetic inductionThe generation of an EMF (and hence currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). in a closed circuit) when the magnetic fluxThe product of magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³. and the area perpendicular to the field. Measured in weber (Wb). linkage through a conductor changes..
The magnitude of the induced EMF is equal to the rate of change of flux linkage: ε = NΔΦ/Δt.
Q2. State Lenz's lawThe direction of an induced EMF is such that it opposes the change in magnetic flux that produced it. This is a consequence of conservation of energyThe capacity to do work. Measured in joules (J)...
The induced EMF always acts in a direction to oppose the change in flux linkage that is producing it.
Q3. Why is there a minus sign in ε = −NΔΦ/Δt?
  • The minus sign represents Lenz's lawThe direction of an induced EMF is such that it opposes the change in magnetic flux that produced it. This is a consequence of conservation of energy..
  • It shows that the induced EMF opposes the change in flux linkage.
  • If the induced EMF aided the change, energy would be created from nothing, violating conservation of energyEnergy cannot be created or destroyed, only transferred from one form to another. The total energy of a closed system remains constant..
Q4. State four methods of inducing an EMF.
  • Moving a wire through a magnetic field, moving a magnet into or out of a coil, changing the currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). in a nearby coil (mutual induction), rotating a coil in a magnetic field.
  • All work by changing the flux linkage.
Q5. Derive the expression ε = BLv for a straight conductor moving through a field.
  • In time Δt, the wire sweeps out area ΔA = LvΔt.
  • Change in flux: ΔΦ = BΔA = BLvΔt.
  • By Faraday's lawThe magnitude of the induced EMF is proportional to the rate of change of magnetic flux linkage.: ε = ΔΦ/Δt = BLvΔt/Δt = BLv.