A rotating coil produces a sinusoidal EMF: $\varepsilon = BAN\omega sin(\omegat)$

Electromagnetic Induction — AQA A-Level Physics

A coil of N turns and area A rotating at angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. ω in a uniform magnetic field B has a 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). that varies with time.
The 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). depends on the angle between the normal to the coil and the field direction. As the coil rotates, this angle changes continuously.

N\Phi = BAN\cos(\omega t)

$NΦ$: 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). (Wb turns)
$B$: magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³.The strength of a magnetic field. The force per unit length per unit 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). on a 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).-carrying conductor perpendicular to the field. Measured in teslaThe SI unit of magnetic flux density. One tesla is the flux density when a force of 1 N acts on a 1 m conductor carrying 1 A perpendicular to the field. (T). (T)
$A$: area of the coil (m²)
$N$: number of turns
$ω$: angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. (rad s⁻¹)
$t$: time (s)

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 the rate of change of this flux linkage. Differentiating the cosine gives a sine:

\varepsilon = BAN\omega\sin(\omega t)

$ε$: instantaneous induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). (V)
$B$: magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³.The strength of a magnetic field. The force per unit length per unit 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). on a current-carrying conductor perpendicular to the field. Measured in teslaThe SI unit of magnetic flux density. One tesla is the flux density when a force of 1 N acts on a 1 m conductor carrying 1 A perpendicular to the field. (T). (T)
$A$: area of coil (m²)
$N$: number of turns
$ω$: angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. (rad s⁻¹)
$t$: time (s)

The peak (maximum) EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). occurs when sin(ωt) = 1:

\varepsilon_0 = BAN\omega

$ε₀$: peak EMF (V)
$B$: magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³.The strength of a magnetic field. The force per unit length per unit current on a current-carrying conductor perpendicular to the field. Measured in teslaThe SI unit of magnetic flux density. One tesla is the flux density when a force of 1 N acts on a 1 m conductor carrying 1 A perpendicular to the field. (T). (T)
$A$: area of coil (m²)
$N$: number of turns
$ω$: angular velocity (rad s⁻¹)

This is the principle behind every AC generator. The coil rotates, flux linkage oscillates as a cosine, EMF oscillates as a sine.
Increasing any of B, A, N, or ω will increase the peak EMF. Doubling the rotation speed doubles ε₀.

Worked Example

A rectangular coil of 80 turns has dimensions 0.12 m × 0.08 m and rotates at 50 revolutions per second in a uniform magnetic field of flux density 0.36 T. Calculate the peak EMF.

Show Solution

List known values

Turns: $N = 80$
Area: $A = 0.12 \times 0.08 = 9.6 \times 10^{-3} \text{ m}^2$
FrequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz).: $f = 50 \text{ Hz}$
Flux density: $B = 0.36 \text{ T}$

Convert frequency to angular velocity

$$\omega = 2\pi f = 2\pi \times 50 = 100\pi \text{ rad s}^{-1}$$

Apply the peak EMF equation

$$\varepsilon_0 = BAN\omega$$ $$= 0.36 \times 9.6 \times 10^{-3} \times 80 \times 100\pi$$

Evaluate

$$\varepsilon_0 = 0.36 \times 9.6 \times 10^{-3} \times 80 \times 314.2$$ $$= 0.36 \times 9.6 \times 10^{-3} \times 25\,136$$ $$= 86.9 \text{ V}$$ $$\approx 87 \text{ V (2 s.f.)}$$

Answer

$\varepsilon_0 = 87$ V

Common Mistake MEDIUM

Students often: Confusing frequency (in Hz) with angular velocity (in rad s⁻¹) when substituting into $\varepsilon_{0} = BAN\omega$.
Instead: ω must be in rad s⁻¹. If you're given frequency f in Hz (or revolutions per second), convert first: $\omega = 2\pi f. Forgetting$ the 2π gives an answer that's 6.28 times too small.