Theory

Electromagnetic induction

How a change in the magnetic flux linked with a circuit generates an electromotive force, how to compute the emf of a moving rod and of a rotating loop, what inductance and the RL circuit are, and where the energy of the magnetic field resides.

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Flux linkage and the Faraday-Neumann-Lenz law
Induced current and Lenz’s law

Flux linkage and the Faraday-Neumann-Lenz law

Throughout this topic the circuits are in vacuum (or in air), and the currents vary slowly enough that the laws of direct-current circuits can be applied at every instant (quasi-static regime).

1. Magnetic flux linkage

Let $γ$ be an oriented closed circuit and let $S$ be a surface whose boundary is $γ$ , with the normal $\overset{n}{^}$ related to the orientation of $γ$ by the right-hand rule. The magnetic flux linked with $γ$ is

Φ_{B} = \iint_{S} B \cdot \overset{n}{^} d S .

The flux of $B$ through a closed surface is zero, by Gauss’s law for magnetism (see the topic on sources of the magnetic field); therefore $Φ_{B}$ is the same for all surfaces whose boundary is $γ$ and depends only on the circuit. The unit of measurement is the weber, $1 Wb = 1 T \cdot m^{2}$ . For a coil of $N$ identical turns the flux linkage is $N$ times the flux through one turn; in a uniform field, with flat turns of area $A$ , it equals $Φ_{B} = N B A cos θ$ .

Flat loop seen edge-on in a uniform field

B

: the flux linkage is

B A cos θ

, where

θ

is the angle between

B

and the normal

\overset{n}{^}

2. Faraday-Neumann-Lenz law

This is an experimental law. When the magnetic flux linked with a circuit varies in time, an induced electromotive force is generated in the circuit,

E = - \frac{d Φ _{B}}{d t},

where $E$ is positive if it acts in the direction of the circuit associated with the normal $\overset{n}{^}$ by the right-hand rule. The emf is determined by the rate at which the flux changes, whatever the value of the flux; the minus sign expresses Lenz’s law on the direction of the induced current. If the flux varies uniformly during the interval $Δ t$ , the emf equals $- \frac{Δ Φ _{B}}{Δ t}$ ; in general this ratio is the average emf.

For example, if the field perpendicular to a coil of $100$ turns with area $20 c m^{2}$ increases uniformly from $0$ to $0.60 T$ in $0.15 s$ , the flux linkage increases by $100 \cdot 20 \cdot 1 0^{- 4} \cdot 0.60 Wb = 0.12 Wb$ and the magnitude of the emf is $\frac{0.12}{0.15} V = 0.80 V$ .

Ways to change the flux

For a flat loop in a uniform field $Φ_{B} = B A cos θ$ , so the flux varies if at least one of the three factors changes.

The magnitude of the field varies, for example because the current that generates it varies or because a magnet approaches; the loop is at rest.
The area of the loop varies, as when a rod slides on two conducting rails; one then speaks of motional emf.
The angle $θ$ varies, as in a loop rotating in the field; this is the principle of the alternator.

The Faraday-Neumann-Lenz law describes all three cases with the same formula. In the first case the emf is due to an induced electric field, in the other two to the Lorentz force on the charges of the moving conductor.

Induced current and Lenz’s law

1. Induced current

If the circuit in which the induced emf $E$ is generated is closed and has total resistance $R$ , and if the self-induction of the circuit is negligible, the induced current

i = \frac{E}{R} = - \frac{1}{R} \frac{d Φ _{B}}{d t}

flows in it. The induced current exists only as long as the flux varies. If the circuit is open, a potential difference equal in magnitude to $E$ is measured across the gap, while the current is zero.

2. Induced charge (Felici's law)

Consider a circuit of resistance $R$ , with negligible self-induction, whose flux linkage changes from the value $Φ_{1}$ at time $t_{1}$ to the value $Φ_{2}$ at time $t_{2}$ . The charge that passes through a cross-section of the circuit during the interval $[t_{1}, t_{2}]$ is

q = \int_{t_{1}}^{t_{2}} i d t = - \frac{Φ _{2} - Φ _{1}}{R} .

The charge depends only on the change in flux, whatever the rate at which it occurs, and for this reason a measurement of charge makes it possible to measure a magnetic field. If a circular loop of radius $5.0 cm$ and resistance $0.20 Ω$ is pulled out of a perpendicular field of $0.40 T$ , the charge that passes through it has magnitude $\frac{0.40 \cdot π \cdot 0.05 0 ^{2}}{0.20} C \approx 16 mC$ , whether the extraction is slow or fast.

Proof

By the definition of induced current $i = - \frac{1}{R} \frac{d Φ _{B}}{d t}$ , and $R$ is constant. Integrating between $t_{1}$ and $t_{2}$ , by the fundamental theorem of calculus we obtain

q = - \frac{1}{R} \int_{t_{1}}^{t_{2}} \frac{d Φ _{B}}{d t} d t = - \frac{Φ _{2} - Φ _{1}}{R} .

Direction of the induced current (Lenz's law)

Because of the minus sign in the Faraday-Neumann-Lenz law, the induced current flows in the direction in which the magnetic field it generates opposes the change in the flux that produced it. In practice one proceeds in three steps: determine whether the flux through the circuit increases or decreases; deduce the direction of the induced field inside the circuit, opposite to the external field if the flux increases and in the same direction if it decreases; obtain the direction of the current with the right-hand rule.

Lenz’s law is a consequence of the conservation of energy. If the induced current reinforced the change in flux, the change would grow by itself and the circuit would produce electrical energy without receiving work from outside.

The external field

B

, pointing out of the page, increases: the induced current flows clockwise and generates inside the loop the field

B^{'}

pointing into the page, which opposes the increase in flux

Warning

The induced field is opposite to the change in the flux, and therefore it is in the same direction as the external field when the latter decreases. If the outward field in the figure decreased, the outward flux would decrease; the induced current would flow counterclockwise and would generate an outward field, in the same direction as $B$ , which partly compensates for the decrease.

The theory continues with 8 more sections (motional electromotive force, rod on rails forces and energy balance, rotating loop and alternator…), plus all worked examples, the solutions and the PDF.

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