Physics

Electromagnetic Induction

Magnetic flux, Faraday's and Lenz's laws, electromotive force, and applications like alternators and transformers.

The magnetic flux ϕ\phi through a surface S is defined as the product of the magnetic field strength and the area of the surface perpendicular to the field.
ϕ=B⃗⋅S⃗=B⋅S⋅cos⁡θ\phi = \vec{\textbf{B}} \cdot \vec{\textbf{S}} = B \cdot S \cdot \cos\theta
Where θ\theta is the angle between the magnetic field vector B and the area vector S (normal to the surface).
The SI unit is the Weber: 1 Wb=1 T⋅m21 \text{ Wb} = 1 \text{ T} \cdot \text{m}^2
Magnetic flux through a surface
The effect of electromagnetic induction is summarized as follows:
An electric current is induced in a circuit if it is exposed to a changing magnetic flux.
Magnetic induction in a loop
Lenz's Law: The direction of the induced current is such that it opposes the change in magnetic flux that produced it.
Faraday's Law of Induction: The induced current is driven by an induced electromotive force (EMF) that is directly proportional to the rate of change of magnetic flux and the number of turns in the coil:
ε=−Ndϕdt\varepsilon = -N \frac{d\phi}{dt}
As we know, for any circuit obeying Ohm's law, ε=I⋅R\varepsilon = I \cdot R, where R is the ohmic resistance of the circuit. The induced current intensity will then be:
I=−NRdϕdtI = -\frac{N}{R} \frac{d\phi}{dt}
From these laws, it follows that an induced EMF will be generated whenever the flux changes, which can happen by changing the magnetic field, the surface area, or the relative angle between the area vector and the magnetic field.
Motional EMF
Consider a conducting rod of length L moving through a magnetic field on a U-shaped conductor. The area of this circuit is variable: S=L⋅v⋅tS = L \cdot v \cdot t
And therefore, the magnetic flux passing through it is also variable:
ϕ=B⋅S⋅cos⁡θ=−B⋅L⋅v⋅t\phi = B \cdot S \cdot \cos\theta = -B \cdot L \cdot v \cdot t
Assuming θ=180∘\theta = 180^\circ, the induced electromotive force (EMF) will be:
ε=−dϕdt=B⋅L⋅v\varepsilon = -\frac{d\phi}{dt} = B \cdot L \cdot v
Motional EMF in a magnetic field
AC Generator, Alternator
Consider a wire loop of area S rotating in a uniform magnetic field B with an angular velocity ω\omega. The flux through the loop is: ϕ=B⋅S⋅cos⁡θ\phi = B \cdot S \cdot \cos\theta
Diagram of an AC generator
MikeRun, CC BY-SA 4.0, via Wikimedia Commons
Where the angle θ\theta varies over time as θ=ωt\theta = \omega t. Thus, the variable flux through the loop is: ϕ=B⋅S⋅cos⁡(ω⋅t)\phi = B \cdot S \cdot \cos(\omega \cdot t)
The induced EMF in N turns is:
ε=−Ndϕdt=N⋅B⋅S⋅ω⋅sin⁡(ω⋅t)\varepsilon = -N \frac{d\phi}{dt} = N \cdot B \cdot S \cdot \omega \cdot \sin(\omega \cdot t)
In the ε\varepsilon vs tt graph, we can see that ε\varepsilon is a periodic sinusoidal function with period T. Where ω=2πf  ⟹  T=2πω\omega = 2\pi f \implies T = \frac{2\pi}{\omega}
In the United States, household AC current operates at f = 60 Hz and an εmax≈170\varepsilon_{\text{max}} \approx 170 V.
Graph of alternating electromotive force
Self-Inductance L
A coil consisting of N turns, with cross-sectional area S and length l, has a self-inductance of:
L=μ⋅N2⋅SlL = \frac{\mu \cdot N^2 \cdot S}{l}
When a varying current flows through the coil, a back EMF is generated:
ε=−dϕdt=−LdIdt\varepsilon = -\frac{d\phi}{dt} = -L \frac{dI}{dt}
Mutual Induction. Transformer.
Used to step up or step down voltage. It consists of two coils (primary and secondary) wound around a laminated iron core. The changing flux generated in the primary induces an EMF in the secondary, related by:
V1V2=N1N2=I2I1derived from assuming P1=P2\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1} \quad \text{derived from assuming } P_1 = P_2
Diagram of a transformer
BillC at the English-language Wikipedia, CC BY-SA 3.0, via Wikimedia Commons