Faraday's Law of Electromagnetic Induction
You already know the story of
electromagnetic induction: move a
magnet and a coil relative to each other and a voltage appears; move them faster, or add more turns,
and the voltage grows; and whatever is induced always opposes the change that made
it. That is the qualitative picture — the flicking galvanometer needle, the bike dynamo, the power
station.
This page makes it quantitative. We will put a single number on "how much field
threads the coil" (the magnetic flux), a second number for "how much when you count
every turn" (the flux linkage), and then state the one law that governs the whole
subject — Faraday's law: the induced emf equals the rate of change of flux
linkage. Every result you met before will drop straight out of a single equation, and you will be
able to calculate the volts, not just say "bigger" or "smaller".
Faraday could see physics — he imagined space filled with invisible lines of force — but
he had almost no formal mathematics. It fell to James Clerk
Maxwell to translate Faraday's field pictures into four compact equations binding
electricity, magnetism and light into a single theory. Maxwell modestly said he had merely put
Faraday's ideas "into mathematical form" — and in doing so predicted that light itself is an
electromagnetic wave.
Magnetic flux: how much field threads the loop
Picture a flat loop of wire held in a
magnetic field of flux density
B (measured in tesla, \text{T}). If the field
passes straight through the loop — perpendicular to its face — the amount of field
"captured" by the loop is the magnetic flux \Phi: the
field strength multiplied by the area it passes through.
\Phi = BA.
Flux is measured in webers (\text{Wb}), and one weber is
one tesla-metre-squared: 1\ \text{Wb} = 1\ \text{T}\,\text{m}^2. Think of
the field lines from the
field-lines picture as
threads: a strong field (B large) or a big loop (A
large) catches more threads, so more flux passes through.
But the loop need not face the field square-on. If the loop is tilted so that its
normal (the line sticking straight out of its face) makes an angle
\theta with the field, only the component of the field along the
normal threads the loop, and the flux is
\Phi = BA\cos\theta.
Two sanity checks. When the field is perpendicular to the loop's face, the normal points
along the field, \theta = 0, \cos 0 = 1,
and we recover \Phi = BA — the maximum. When the field lies flat
in the plane of the loop, the normal is at right angles to the field,
\theta = 90^\circ, \cos 90^\circ = 0, and no
field threads the loop at all: \Phi = 0.
Worked example — computing the flux
A rectangular coil has area A = 2.0\times10^{-2}\ \text{m}^2
(200\ \text{cm}^2) and sits in a uniform field of
B = 0.15\ \text{T}.
-
Face-on to the field (\theta = 0):
\Phi = BA = 0.15 \times 2.0\times10^{-2} = 3.0\times10^{-3}\ \text{Wb} = 3.0\ \text{mWb}.
-
Tilted so the normal is at 60^\circ to the field:
\Phi = BA\cos 60^\circ = 3.0\times10^{-3} \times 0.5 = 1.5\times10^{-3}\ \text{Wb} = 1.5\ \text{mWb}.
Tilting the coil halves the flux.
Flux linkage: counting every turn
A real coil is not one loop but many — a solenoid might have hundreds. Every one of its
N turns is threaded by the same flux \Phi, and
each turn contributes its own induced emf, so what matters for induction is the flux multiplied by the
number of turns. This product is the flux linkage:
\text{flux linkage} = N\Phi = NBA\cos\theta.
Its unit is the weber-turn (often just written \text{Wb}).
A single flat loop has flux linkage equal to its flux; a 500-turn coil in the same field links 500
times as much. Flux linkage is the quantity Faraday's law actually cares about — miss the factor of
N and every induced-emf answer comes out too small by that same factor.
Worked example — flux linkage
Wind N = 500 turns onto the same coil
(A = 2.0\times10^{-2}\ \text{m}^2) and place it face-on in the
0.15\ \text{T} field. The flux through each turn is still
\Phi = 3.0\times10^{-3}\ \text{Wb}, so the flux linkage is
N\Phi = 500 \times 3.0\times10^{-3} = 1.5\ \text{Wb (weber-turns)}.
Faraday's law: emf is the rate of change of flux linkage
Now the central law. Faraday found that the induced emf \varepsilon is
exactly the rate of change of flux linkage — how fast
N\Phi is changing, measured in weber-turns per second (which are
volts):
\varepsilon = -\frac{\mathrm{d}(N\Phi)}{\mathrm{d}t}.
Read it carefully. It is the rate of change that induces an emf — not the size of
the flux itself. A coil sitting in an enormous but steady flux has
\frac{\mathrm{d}(N\Phi)}{\mathrm{d}t} = 0 and induces nothing.
A tiny flux that changes quickly can induce far more. Everything you learned qualitatively is encoded
here: a bigger N, a faster change, a stronger magnet — each raises the rate
of change and so raises \varepsilon.
When the change is steady we can replace the derivative with an average, using the change
\Delta over a time \Delta t:
\varepsilon = -\frac{\Delta(N\Phi)}{\Delta t} = -\frac{N\,\Delta\Phi}{\Delta t}.
The minus sign is Lenz's law, coming next — for the size of the emf you
take the magnitude and worry about direction separately.
Worked example — induced emf from a collapsing field
The field through the 500-turn coil above
(A = 2.0\times10^{-2}\ \text{m}^2, face-on) is switched off, falling
steadily from B = 0.15\ \text{T} to zero in
\Delta t = 0.10\ \text{s}. Find the induced emf.
-
Flux linkage at the start: N\Phi = 1.5\ \text{Wb} (from above);
at the end it is 0. So
\Delta(N\Phi) = 0 - 1.5 = -1.5\ \text{Wb}.
-
Apply Faraday's law:
\varepsilon = -\frac{\Delta(N\Phi)}{\Delta t} = -\frac{-1.5}{0.10} = +15\ \text{V}.
The induced emf has magnitude 15\ \text{V}. Halve the switch-off time to
0.05\ \text{s} and the emf doubles to
30\ \text{V} — same change of flux, packed into less time.
Lenz's law: nature pushes back
Which way does the induced current flow? Lenz's law answers with the minus sign in
Faraday's law: the induced emf (and the current it drives) always acts to oppose the change
of flux that produced it. Push a magnet's north pole into a coil and the near face of the
coil becomes a north pole too, shoving back against you; pull it out and that face turns south, trying
to drag the magnet back in. Either way the coil resists what you are doing to it.
This is not nature being contrary — it is conservation of energy. Because the coil
resists, you must do mechanical work to keep the flux changing, and that work is exactly the
electrical energy the coil delivers. If the induced current helped the change instead, the
magnet would accelerate itself and pour out free energy forever — a perpetual-motion machine. The
minus sign is the universe forbidding that.
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Magnetic flux through a loop of area A in a field
B is \Phi = BA\cos\theta, where
\theta is the angle between the field and the loop's normal;
for an N-turn coil the flux linkage is
N\Phi (unit: weber-turn).
-
Faraday's law: the induced emf equals the rate of change of flux linkage,
\varepsilon = -\dfrac{\mathrm{d}(N\Phi)}{\mathrm{d}t}. Only a
changing flux linkage induces an emf; a steady one, however large, induces nothing.
-
Lenz's law: the induced emf and current always oppose the change producing them
— the meaning of the minus sign. This is a statement of energy conservation.
-
It is the RATE of change of flux that matters, not the flux. A coil bathed in a
huge but constant flux induces nothing — \frac{\mathrm{d}(N\Phi)}{\mathrm{d}t} = 0.
Only while the flux is changing is there an emf. Students who plug the flux itself into
Faraday's law get zero volts wrong every time.
-
The minus sign (Lenz) means the effect OPPOSES the change. Push a magnet in and
the coil pushes back; the induced current never helps the motion along. If your reasoning has the
coil assisting the change, you have built a perpetual-motion machine — check the sign.
-
Flux linkage includes the number of turns. Faraday's law uses
N\Phi, not \Phi. Forgetting the
N divides every emf by the number of turns — often a factor of hundreds.
Also mind \theta: it is measured from the normal to the coil,
so "field along the coil face" means \theta = 90^\circ and zero flux,
not maximum.
Changing the flux: the a.c. generator
Faraday's law says an emf appears whenever N\Phi changes, and there are
three everyday ways to make it change: move a magnet in or out of a coil (the field through the coil
grows and shrinks); move a wire across a field (coming next); or spin a coil in a
steady field. The spinning coil is the heart of every generator.
As a coil rotates, the angle \theta between the field and its normal sweeps
round, so the flux linkage is N\Phi = NBA\cos(\omega t) — it swings
smoothly between +NBA and -NBA. Its rate of
change, and hence the induced emf, is
\varepsilon = NBA\,\omega\sin(\omega t): a sinusoidal a.c.
voltage. That is why mains electricity is alternating — it falls straight out of a coil going
round and round.
The most-tested fact hides in that gradient. The emf is the slope of the flux-linkage graph,
so it is largest when the flux is momentarily zero (the coil's plane lies along the
field, sweeping across the lines fastest) and zero when the flux is largest (the coil
faces the field square-on, where \cos is flat and, for an instant, the flux
is not changing at all). Peak flux, zero emf; zero flux, peak emf — a quarter-turn out of step.
In the graph below, choose how the flux through one turn changes with time and watch the induced emf —
the coil's number of turns scales the emf but never the flux. Notice the emf is always
-N times the gradient of the flux curve: a steady rise gives a
constant emf, a triangular flux gives a square-wave emf, and a sinusoidal flux gives a sinusoidal emf
a quarter-cycle ahead.
A rod on rails: motional emf, \varepsilon = BLv
There is a fourth way to change flux that gives a specially clean formula. Slide a straight conducting
rod of length L along two rails at speed v,
through a field B pointing at right angles to the whole arrangement. As the
rod moves, the circuit it closes off grows in area, so the flux through that circuit
keeps increasing — and a changing flux means an induced emf.
We can get the formula straight from Faraday's law. In a time \Delta t the
rod moves a distance v\,\Delta t, sweeping out extra area
\Delta A = L\,(v\,\Delta t). The extra flux is
\Delta\Phi = B\,\Delta A = BLv\,\Delta t, so (with one turn,
N = 1) the induced emf has magnitude
\varepsilon = \frac{\Delta\Phi}{\Delta t} = \frac{BLv\,\Delta t}{\Delta t} = BLv.
Clean and memorable: a rod of length L cutting field lines at speed
v generates \varepsilon = BLv. Faster, longer, or
a stronger field — each raises the emf, exactly as the general law demands. (If the rod moves at an
angle, only the component of velocity perpendicular to the field counts.)
Worked example — a rod cutting field lines
A metal rod of length L = 0.25\ \text{m} slides at
v = 4.0\ \text{m s}^{-1} at right angles through a field
B = 0.30\ \text{T}.
\varepsilon = BLv = 0.30 \times 0.25 \times 4.0 = 0.30\ \text{V}.
A modest third of a volt — but a wingtip on an airliner is tens of metres long and moves at
hundreds of metres per second through the Earth's field, so a real emf of a volt or two appears
across the wings in flight. Nothing to power a light with (there is no closed circuit), but real and
measurable.
Drop a small strong magnet down a vertical copper pipe and it does not clatter to
the bottom — it drifts down eerily slowly, as if falling through honey, even though copper is not
magnetic. Faraday and Lenz explain it perfectly. As the magnet falls, the flux through each ring of
the pipe around it changes, inducing little circulating currents in the copper — called
eddy currents. By Lenz's law those currents oppose the change, so they act to slow
the magnet's fall: the ring just below is pushed away and the ring just above pulls back. The
magnet's gravitational energy is quietly turned into heat in the copper, and it settles to a gentle
steady speed.
The very same physics runs the modern world. A power station is, at its heart, just
a magnet spun inside coils — coal, gas, nuclear, hydro and wind all differ only in what does the
spinning. An induction hob uses a rapidly changing field to induce heating eddy
currents directly in the base of the pan (the hob top stays cool). A wireless phone
charger induces a current in a coil in the phone from a changing field in the pad, with no
wires touching. Eddy-current brakes on trains and roller-coasters slow the car with
no pads to wear out. One law of Faraday's, discovered in 1831, quietly powering the lot.