Faraday's Laws of Electromagnetic Induction

Electromagnetic induction

• The phenomenon of producing an induced emf in a conductor or conducting coil due to changing magnetic flux is called electromagnetic induction.
• Faraday enunciated two laws on the basis of the experimental evidence collected in 12.1.

Faraday's first law

• Whenever there is a change of magnetic flux in a closed circuit, an induced emf is produced in the circuit. Also, if a conductor cuts the lines of magnetic field, an emf is induced between its ends.
• This is a qualitative law: it only tells us that an emf appears and under what circumstances, not how large it is.

Faraday's second law

• The magnitude of the induced emf produced in the circuit is directly proportional to the rate of change of magnetic flux linked with the circuit.
• This is the quantitative law: it fixes the magnitude of the induced emf.

The mathematical form

• If $\phi$ is the magnetic flux linked with the coil at any instant $t$, then

$$e \propto \dfrac{d\phi}{dt}$$

$$e = K\dfrac{d\phi}{dt}$$

• where $K$ is the constant of proportionality. If $e$, $\phi$ and $t$ are measured in the same system of units, $K = 1$:

$$e = \dfrac{d\phi}{dt}$$

• Combining this with Lenz's law (which fixes the direction, next section) gives the standard form

$$e = -\dfrac{d\phi}{dt}$$

• If $\phi'$ is the flux associated with a single turn, then for a coil of $n$ turns the total flux linked is $\phi = n\phi'$, so

$$e = -n\dfrac{d\phi'}{dt}$$

• This is also known as the flux rule: the emf is equal to the rate at which the magnetic flux through a conducting circuit is changing.

Units and a physical remark

• In SI units $e$ is measured in volt and $\dfrac{d\phi}{dt}$ in weber per second. Flux is in weber, time in second.
• From the magnetic effect of current we already knew that charges in motion can exert a force or torque on a stationary magnet. Faraday's experiments show the converse: a bar magnet in motion (a time-varying magnetic field) exerts a force on the stationary charges inside a conductor, and so causes an induced emf across the ends of an open conductor, or an induced current in a closed circuit.

Standard results

• $e = -\dfrac{d\phi}{dt}$ for one turn; $e = -N\dfrac{d\phi}{dt}$ for $N$ turns.
• For a uniform change over a time $\Delta t$: $\lvert e\rvert = \dfrac{N(\phi_{\text{final}}-\phi_{\text{initial}})}{\Delta t} = NA\dfrac{\Delta B}{\Delta t}$ for a fixed area normal to $\vec{B}$.
• Induced current $I = \dfrac{\lvert e\rvert}{R}$ in a closed circuit of resistance $R$.