So far a magnetic field has always pushed a moving charge. Now flip the whole idea around. Take a plain metal rod — no battery, nothing connected — and simply move it through a magnetic field. Something remarkable happens: a voltage appears across its ends, out of nowhere. Move it faster and the voltage grows. This is electromagnetic induction, the trick behind every power station on Earth, and it all starts with charges inside the moving rod feeling a familiar force.
📘 What you need to know
Electromagnetic induction is when an e.m.f. is induced by relative movement between a conductor and a magnetic field
It happens when a conductor moves relative to a field, or the field changes relative to the conductor
As the conductor cuts field lines, free electrons feel a magnetic force and are pushed to one end
This separates charge, creating a potential difference — an induced e.m.f.
The induced e.m.f. is the work done per unit charge in separating the charges to the ends of a conductor
For a straight rod: ε = BLv (and for a coil of N turns, ε = BLvN)
The e.m.f. is bigger if the field is stronger, the conductor is longer, or it moves faster (or has more turns)
The rod must move perpendicular to the field so it actually cuts the lines
Connect the ends to a closed circuit and the induced e.m.f. drives an induced current
Why does a voltage appear?
The rod is full of free electrons. When you move the whole rod sideways through the field, you drag those electrons along with it — and a moving charge in a magnetic field feels a force (the old F = Bqv from earlier pages). That force pushes the electrons along the rod, all toward the same end.
The free electrons feel a magnetic force and drift to one end of the rod
That end becomes negative; the other end, left short of electrons, becomes positive
The charges are now separated, so a potential difference sits across the rod
This p.d. is the induced e.m.f. — and the work needed to separate the charges came from you, pushing the rod
Push the rod through the field and its free electrons pile up at one end. That charge separation is the induced e.m.f. — a battery made from nothing but motion.
Notice there’s no cell anywhere. The energy for this voltage isn’t stored in the rod — it comes from the work you do pushing the rod against the magnetic force on those electrons. That’s why “induced e.m.f.” is defined as the work done per unit charge in separating the charges. Move the rod, do work, get a voltage. Stop moving it, and the voltage vanishes.
The size of the induced e.m.f.
For a straight conductor cutting perpendicular field lines, the induced e.m.f. is refreshingly simple.
Induced e.m.f. in a moving rodε = BLvε = e.m.f. (V) • B = flux density (T) • L = length in the field (m) • v = speed (m s−1)
Coil the wire into N loops and every loop adds its own e.m.f., so they stack up:
Induced e.m.f. in a coil of N turnsε = BLvNN = number of turns on the coil
Read straight off the equation and you get the three (or four) ways to boost the voltage:
Increase…
Effect on e.m.f.
Why
Field strength B
Bigger e.m.f.
Denser field, more lines cut per second
Length L in the field
Bigger e.m.f.
More conductor to separate charge along
Speed v
Bigger e.m.f.
Field lines cut faster
Turns N
Bigger e.m.f.
Each loop adds its own e.m.f.
Move the rod
force on electrons
Charge separates
p.d. across the ends
e.m.f. ε = BLv
WE 1
A straight metal rod of length 40 cm moves at 6.0 m s−1 at right angles through a uniform magnetic field of flux density 0.25 T. (a) Calculate the e.m.f. induced across its ends. (b) State what happens to the e.m.f. if the rod is moved twice as fast. (c) State what happens to the e.m.f. if the rod is instead moved along the field lines.
(a) Step 1 — use the induced-e.m.f. equationε = BLv = (0.25)(0.40)(6.0)ε = 0.60 V(b) double the speedε ∝ v, so doubling v doubles εε = 1.2 V(c) moving along the field lines
The rod no longer cuts any field lines.
ε = 0 — no e.m.f. is induced at allRemember to convert 40 cm to 0.40 m first. And part (c) is the classic trap: the rod has to move across the field to cut lines. Slide it along the lines and nothing happens, however fast you go.
Seeing it happen
Two simple demonstrations show induction in action. In both, a voltmeter reads zero until something moves — the whole effect lives in the movement.
Experiment 1: a magnet through a coil
Connect a coil to a sensitive voltmeter and push a bar magnet in and out.
Push the magnet in and the needle flicks one way; pull it out and it flicks the other. Hold it still — inside or outside — and the reading drops to zero.
What you see, step by step:
Magnet still: no change, no cutting of field lines, so no e.m.f. — the needle sits at zero
Magnet moving in: its field lines sweep through the coil, the flux changes, and an e.m.f. is induced
Magnet pulled out: the flux now changes the other way, so the e.m.f. — and the needle — reverses
Move it faster: the flux changes faster, so the induced e.m.f. is bigger
Experiment 2: a wire between two magnets
Same idea, other way round: keep the field still and move a wire through it instead. Connect a long wire to a voltmeter and sweep it between the poles of two magnets. Exactly the same rules apply.
Wire still: no field lines cut, no e.m.f.
Wire moving through the field: it cuts field lines, the flux changes, and an e.m.f. is induced
Wire moved back out: the e.m.f. reverses sign
Bigger e.m.f. from a longer wire, a faster sweep, or stronger magnets
The lovely symmetry here: it does not matter whether you move the magnet or move the wire — only the relative motion counts. That’s the heart of the definition: induction needs relative movement between the conductor and the field. From the physics’ point of view, “magnet moving past a still wire” and “wire moving past a still magnet” are the same event.
WE 2
A coil of 50 turns, each of effective length 0.25 m in the field, moves through a uniform magnetic field of flux density 0.15 T. The e.m.f. induced across the coil is 1.5 V. Calculate the speed at which the coil is moving.
Step 1 — use the coil equation and rearrange for vε = BLvN → v = ε / (BLN)Step 2 — substitutev = 1.5 / [(0.15)(0.25)(50)]v = 1.5 / 1.875v = 0.80 m s⁻¹Don’t forget the N. With 50 turns each loop chips in, so you only need a gentle 0.8 m s⁻¹ to reach 1.5 V. Drop the N by mistake and you’d get a speed 50× too big.
WE 3
An aircraft with a wingspan of 36 m flies horizontally at 250 m s−1 through the vertical component of the Earth’s magnetic field, which has a flux density of 5.0 × 10−5 T. (a) Calculate the e.m.f. induced across the wingtips. (b) Explain why this e.m.f. cannot be used to power anything on the plane through a simple circuit joining the wingtips.
(a) Step 1 — the wing is a moving rodε = BLv = (5.0 × 10⁻⁵)(36)(250)ε = 0.45 V(b) why no useful current?
A wire joining the tips would move through the same field at the same speed.
The same e.m.f. is induced in it too, in the opposite sense.
the two e.m.f.s cancel, so no current flows round the loopA real number from real physics: a big jet really does build about half a volt across its wings. But to get a current, the return path must sit outside the field — if the whole loop moves through the same field together, the induced e.m.f.s cancel and nothing flows.
⚡ Working an induced-e.m.f. question
Straight rod? Use ε = BLv. Coil? Use ε = BLvN.
Convert everything: cm → m, mT → T.
Check the geometry. The rod must move perpendicular to the field to cut lines. Along the field → zero.
Rearranging? Solve for whichever quantity is missing (v = ε/BLN, etc.).
Want a current? The circuit must be closed, with the return path outside the field.
Bigger e.m.f.? Bigger B, L, v or N — straight from the equation.
💡 Top tips
The rod must cut field lines — move it perpendicular to B, not along it.
No movement, no e.m.f. Induction lives entirely in the change.
Only relative motion matters — move the magnet or the conductor, same result.
Convert to SI first: cm → m, mT → T.
For a coil, don’t drop the N — each turn adds its own e.m.f.
For a current to flow, the circuit must be closed outside the field.
⚠ Common mistakes
Thinking a stationary conductor in a steady field has an e.m.f. It needs movement (or a changing field)
Using a rod moving along the field lines — it cuts nothing, so ε = 0
Leaving L in cm or B in mT in ε = BLv
Forgetting the N for a coil
Assuming an induced e.m.f. always drives a current. It only does if the circuit is complete
Thinking the energy comes from nowhere — it comes from the work done moving the conductor
Quick recap:Electromagnetic induction is an e.m.f. induced by relative movement between a conductor and a magnetic field. As the conductor cuts field lines, its free electrons feel a magnetic force and pile up at one end, separating charge and creating a p.d. — the induced e.m.f., defined as the work done per unit charge in separating them. For a rod, ε = BLv; for a coil, ε = BLvN. It grows with a stronger field, longer conductor, faster motion, or more turns, and drives a current only if the circuit is closed outside the field.
We’ve been talking loosely about field lines being “cut”. Time to make that precise. The proper way to count how much field passes through a loop is a quantity called magnetic flux — and an e.m.f. is really induced by a change in flux, not just by motion. Nail that idea and Faraday’s law falls straight out of it. Next page: Magnetic Flux.
Induction not quite clicking?
Book a free meeting and we’ll work through ε = BLv, the charge-separation argument, and why only relative motion matters.
