Faraday's & Lenz's Laws

PHYElectromagnetic Induction

Faraday's law states that an EMF is induced whenever the magnetic flux linked with a circuit changes, with a magnitude equal to the rate of change of flux linkage; Lenz's law fixes its direction so that it opposes the change. Students compute EMF from a changing field $\varepsilon = -N\,d\phi/dt$ and from a flux given as a function of time by differentiating.

This is the conceptual heart of the whole chapter.

Magnetic flux

\phi = B A \cos\theta

B

= field (T),

A

= area (m

^2

\theta

= angle between

B

and the area normal;

\theta = 0^{\circ}

when the plane is perpendicular to

B

Faraday's law (N turns)

\varepsilon = -N\,\dfrac{d\phi}{dt}

N

= turns,

d\phi/dt

= rate of change of flux per turn (Wb/s); the minus sign is Lenz's law.

EMF for a uniform change in B

\varepsilon = -N\,\dfrac{\Delta\phi}{\Delta t} = -N A\,\dfrac{\Delta B}{\Delta t}

used when

B

changes uniformly with

A,\theta

fixed;

\Delta B

= change in field,

\Delta t

= time interval.

Flux changes via THREE routes: changing $B$ , changing area $A$ , or changing orientation $\theta$ — Faraday's law covers all three.
When the coil plane is perpendicular to $B$ , the normal is along $B$ so $\theta = 0^{\circ}$ and $\phi = BA$ (a frequent setup).
For a flux given as $\phi(t)$ (e.g. $\phi = 5t^2 + 3t + 2$ ), the EMF is the time-derivative: $\varepsilon = -N\,d\phi/dt$ evaluated at the required instant.
Lenz's law is energy conservation: the induced current's field opposes the original flux change, so work must be done to maintain the change.
Flux linkage is $N\phi$ ; always multiply by the number of turns when finding the total EMF.
A constant term in a $\phi(t)$ polynomial (e.g. the $+2$ ) contributes nothing to the EMF, since its derivative is zero.
Use SI throughout: $B$ in T, $A$ in m $^2$ , $t$ in s gives $\varepsilon$ in volts directly.

Where the marks go

Forgetting the factor $N$ (turns) when computing the total induced EMF.
Using $\theta$ as the angle between $B$ and the coil PLANE instead of the normal — 'plane perpendicular to $B$ ' means $\theta = 0^{\circ}$ , giving $\cos\theta = 1$ .
For $\phi(t)$ , plugging $t$ into $\phi$ instead of differentiating first, or substituting the time before differentiating.
Dropping the minus sign when direction is asked (Lenz) — keep it to justify the opposing sense, but report magnitude when only the value is needed.

How the board asks it

Numericalflux given as a function of time $\phi(t)$
The magnetic flux linked with a coil of $200$ turns is given by $\phi = 5t^2 + 3t + 2$ (in $Wb$ ). Calculate the magnitude of the induced EMF in the coil at $t = 2\,s$ .
Numericalemf from a uniformly changing field $B$ with the plane perpendicular to $B$
A coil of $50$ turns and area $4 \times 10^{-2}\,m^2$ is held with its plane perpendicular to a magnetic field that falls uniformly from $0.6\,T$ to $0.1\,T$ in $0.2\,s$ . Find the magnitude of the EMF induced in the coil.
Define / statefaraday's laws of electromagnetic induction
State Faraday's laws of electromagnetic induction and write the expression for the EMF induced in a coil of $N$ turns when the flux linked with it changes.
Give reasonslenz's law as a consequence of energy conservation
A bar magnet is pushed with its north pole towards a closed coil. Using Lenz's law, give reasons for the direction of the induced current, and show that Lenz's law is a consequence of the conservation of energy.
Applicationthe three routes ( $B$ , $A$ , $\theta$ ) by which flux changes
Explain how an EMF can be induced in a coil placed in a magnetic field of constant magnitude, naming the two quantities whose change can produce the induced EMF.

Practise this topic

NextMotional EMF

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.