Faraday's Law: the EMF a Changing Flux Induces

Q: Why does moving the magnet faster give a bigger EMF but the same charge?

Because the EMF depends on the rate of flux change and the charge depends only on the total flux change. ε = −N dΦ/dt, so halving the time doubles the EMF; but q = ∫(ε/R) dt = NΔΦ/R, and every dt cancels. The table above shows the EMF changing by a factor of a thousand while the charge sits still.

A magnet sitting still next to a coil does nothing. Move it and the coil pushes a current. This gives you ε = −N dΦ/dt from a flux that changes, the flux linkage before and after, the BLv of a rod sliding on rails, and the charge that flows — which, oddly, does not depend on how fast you do it.

Last updated 11 Nov 2026

Induced EMF from a changing flux, and from a moving rod

ε = −N ΔΦ/Δt · ε = BLv

Number of turns on the coilEvery turn links the same flux, so N turns in series give N times the EMF. This is why a generator or a transformer winding has hundreds of turns rather than one. Cambridge 9702 calls NΦ the flux linkage and examines it under outcome 20.5.3; it is measured in weber-turns.

Area of the coil (cm²)In square centimetres — 25 cm² is a 5 cm by 5 cm loop. Only the area the field actually passes through counts, and only the component of the field perpendicular to the coil. A coil turned edge-on to the field links no flux at all however big it is.

Flux density through the coil at the start (T)May be negative, which means the field points through the coil the other way. For scale: the Earth’s field is about 5 × 10⁻⁵ T, a fridge magnet about 5 × 10⁻³ T at its face, a laboratory magnet a few tenths of a tesla.

Flux density through the coil at the end (T)Set this equal to the starting value and the EMF goes to exactly zero, which is the most important single fact on this page: a field does not induce an EMF, only a change in flux does. A coil parked inside the strongest magnet in the building reads nothing.

Time the change takes (s)Faraday’s law is about the rate. Halve this and the EMF doubles. The charge that flows round the circuit, in the row below, does not change at all — a slower change gives a smaller EMF for a proportionally longer time, and the two cancel exactly.

Speed of the sliding rod (m/s)For the second half of the page. The rod slides on rails inside the starting field B, and the rails are taken to be √A apart, so the circuit is square at the moment the area equals the area you gave above. Its EMF is BLv, which is not a new law — it is −dΦ/dt with Φ = BLx and dx/dt = v.

The same coil and magnet twice: on the left before the change, on the right after it. The number of flux arrows in each half follows that half's field, so you can see the flux fall; they reverse if you make a field negative, and the pole facing the coil swaps with them. The bar underneath is the EMF that results, drawn out to the right for a positive EMF and to the left for a negative one.

3.0000VExample

A 200-turn coil of 25 cm², field falling from 0.40 T to 0.10 T in 0.050 s

One law, two faces

ε = −N dΦ/dt · Φ = BA cosθ · flux linkage = NΦ · sliding rod: ε = BLv · charge: q = NΔΦ/R

Φ: magnetic flux, in webers. Cambridge 9702 outcome 20.5.2 states it as Φ = BA, for a field perpendicular to the area. A weber is a volt-second (SI Brochure, Table 4), so “one weber per second is one volt” is Faraday’s law written in units
the minus sign: Lenz’s law. The induced EMF opposes the change that produced it — CIE words this as “in such a direction as to oppose the change producing it”, outcome 20.5.4. It is a consequence of energy conservation, not an extra assumption: an EMF that helped the change along would build its own energy out of nothing
N: turns. It multiplies the EMF because the turns sit in series and each sees the same dΦ/dt. Be careful which quantity a question is asking for: Φ is per turn, NΦ is the flux linkage, and mixing them up is a factor-of-N error that looks entirely plausible
BLv: the EMF of a rod of length L sliding at speed v across a field B, with B, L and v mutually perpendicular. Not a separate law: the circuit’s area grows at Lv, so the flux grows at BLv, so the EMF is BLv. If the rod slides along the field instead of across it, the area swept links no new flux and the EMF is zero
q = NΔΦ/R: the charge that flows, found by integrating ε/R dt. Every dt cancels. This is the quantity a ballistic galvanometer measures, and it is why a search coil can measure a field without knowing how quickly you yanked it out

Worked example

A 200-turn coil of 25 cm², field falling from 0.40 T to 0.10 T in 0.050 s
AREA INTO SI FIRST. 25 cm² = 25 × 10⁻⁴ m² = 2.5 × 10⁻³ m². Note the square: the factor is 10⁻⁴ and not 10⁻². This is the single most common arithmetic slip on induction questions.
FLUX BEFORE AND AFTER, PER TURN. Φ₁ = B₁A = 0.40 × 2.5 × 10⁻³ = 1.0 × 10⁻³ Wb. Φ₂ = 0.10 × 2.5 × 10⁻³ = 2.5 × 10⁻⁴ Wb. So ΔΦ = −7.5 × 10⁻⁴ Wb — negative, because the flux fell.
APPLY THE LAW. ε = −NΔΦ/Δt = −200 × (−7.5 × 10⁻⁴)/0.050 = +0.15/0.050 = 3.0 V. The two minus signs cancel, which is what a falling flux always does.
SANITY-CHECK THE SIZE. The flux linkage went from 0.20 Wb-turns to 0.05, a drop of 0.15 Wb-turns, in a twentieth of a second. 0.15 weber-turns per 0.05 seconds is 3 weber-turns per second, and a weber per second is a volt. You can do this one in your head once you believe the units.
NOW THE ROD. The rails are √A = 0.050 m apart, and the rod slides at 2.0 m/s in the starting field of 0.40 T: ε = BLv = 0.40 × 0.050 × 2.0 = 0.040 V. Far smaller than the coil's 3 V, because one turn is doing the work of 200 and the area is changing much more gently.
AND THE CHARGE, WHICH IGNORES THE CLOCK. Through a 10 Ω circuit, q = NΔΦ/R = 0.15/10 = 0.015 C. Take ten times as long over the change and the EMF falls to 0.30 V — and exactly 0.015 C still flows. Try it: set the time to 0.50 s and watch the charge row refuse to move.

Same flux change, different timings — at this page’s other defaults

The EMF and the current span a factor of a thousand down this table and the charge does not move at all. That is not a coincidence or an approximation — q = NΔΦ/R contains no time, because the smaller current flows for proportionally longer. It is the reason a search coil can measure a magnetic field without you having to pull it out at any particular speed.
Time for the change	EMF	Current through 10 Ω	Charge that flows
0.005 s	30.0 V	3.00 A	0.015 C
0.050 s	3.00 V	0.300 A	0.015 C
0.500 s	0.300 V	0.0300 A	0.015 C
5.00 s	0.0300 V	0.00300 A	0.015 C

A field does nothing; a changing flux does everything

Nothing about a magnetic field by itself induces an EMF. A coil can sit inside the strongest magnet in the laboratory for a week and read exactly zero. What induces an EMF is a change in the flux through the coil, and there are only three ways to get one: change the field, change the area, or turn the coil. The diagram on this page shows the first. The sliding rod shows the second. A generator does the third, and does it sinusoidally, which is where alternating current comes from.

The minus sign is the part worth arguing about. On its own it is meaningless, because the sign of an EMF depends on which way round the loop you decided to call positive. Fix that first: pick a direction for the coil’s normal, call flux that way positive, and let the positive sense round the loop be the right-handed one about it. Then −N dΦ/dt tells you the EMF in that sense, and this page reports it with that convention. What the minus sign means, physically, is that the induced current’s own field pushes back against the change — which is Lenz’s law, and has its own page at Lenz’s law and the direction of the induced current.

BLv is not a second law to memorise. A rod of length L sliding at v on rails in a field B sweeps out area at a rate Lv, so the flux through the circuit grows at BLv, so the EMF is BLv. It drops out of Faraday’s law in one line. The useful consequence is that if the rod slides along the field rather than across it, the swept area links no new flux and there is no EMF at all, however fast it goes.

Where this goes wrong, most often, is the area. Twenty-five square centimetres is 2.5 × 10⁻³ m², not 0.25 m²; the conversion factor is squared. After that it is the N: a question that gives you the flux per turn and asks for the EMF of the whole coil wants a factor of N that is not in the number you were given. If an induction answer is out by a round factor of a hundred or a few hundred, it is almost always one of those two. For what the coil’s own field does back, see self and mutual inductance, and for the field a current makes in the first place, the field of a wire, coil and solenoid.

Frequently asked questions

Why does moving the magnet faster give a bigger EMF but the same charge?

Because the EMF depends on the rate of flux change and the charge depends only on the total flux change. ε = −N dΦ/dt, so halving the time doubles the EMF; but q = ∫(ε/R) dt = NΔΦ/R, and every dt cancels. The table above shows the EMF changing by a factor of a thousand while the charge sits still.

Can the EMF be negative? This page returns negative numbers.

It can, and the sign tells you which way round the loop the EMF drives. It is only meaningful relative to the positive sense you chose — reverse that choice and every sign on the page reverses. Most exam questions ask only for the magnitude, which is why there is a row above giving it with the sign set aside. The band under the headline says what the sign means here: positive means the flux is falling.

What if the field is not perpendicular to the coil?

Then Φ = BA cosθ, with θ measured between the field and the normal to the coil — not between the field and the coil’s plane. A coil edge-on to the field (θ = 90°) links zero flux. This calculator takes the field as perpendicular, so if your question gives an angle, multiply the fields you type in by cosθ first.

Is BLv a different law from Faraday’s?

No. The circuit’s area grows at Lv as the rod slides, so the flux grows at BLv, so the EMF is BLv. It is Faraday’s law with the area changing instead of the field. The deeper reason it works — that the free charges in the moving rod feel a qvB force along the rod — is the same physics seen from a different frame, and is covered on the magnetic force page.

Why does a transformer not work on direct current?

Because a steady current makes a steady flux, and a steady flux induces nothing. Set the two fields on this page equal and the EMF goes to exactly zero; that is a transformer on DC. It is also why a transformer primary connected to DC usually burns out: with no back-EMF, nothing limits the current but the winding’s own resistance.

Related calculators

References

Annual Syllabus, Class XII, Physics, Code 42, Session 2025-26, Central Board of Secondary Education, edustud.nic.in/edu/SYLLABUS_2025_26/12/12_Physics_EM.pdf (read 11 November 2026). The board’s own document, cited and not endorsed by it. Unit I Chapter 1 asks for “Electric flux, statement of Gauss’s theorem and its applications to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell (field inside and outside)”. Unit III Chapter 4 asks for “force on a moving charge in uniform magnetic and electric fields”, “torque experienced by a current loop in uniform magnetic field” and “Current loop as a magnetic dipole and its magnetic dipole moment”. Unit IV Chapter 6 is one line and covers three of this batch’s records: “Electromagnetic induction; Faraday’s laws, induced EMF and current; Lenz’s Law, Self and mutual induction”. The Hall effect is not in this document.
Syllabus: Cambridge International AS & A Level Physics 9702, for examination in 2025, 2026 and 2027, Cambridge Assessment International Education, cambridgeinternational.org/Images/664565-2025-2027-syllabus.pdf (read 11 November 2026). Cited for outcomes a Cambridge candidate is examined on that the Indian board words differently or omits. 20.3.3: “understand the origin of the Hall voltage and derive and use the expression V_H = BI /(ntq), where t = thickness”. 20.3.4: “understand the use of a Hall probe to measure magnetic flux density”. 20.5.2: “recall and use Φ = BA”. 20.5.3: “understand and use the concept of magnetic flux linkage”. 20.5.4 asks candidates to explain experiments demonstrating “that the induced e.m.f. is in such a direction as to oppose the change producing it”. 20.5.5: “recall and use Faraday’s and Lenz’s laws of electromagnetic induction”. Note that induction is topic 20.5 here and not a topic of its own; topic 21 is alternating currents. The copy read was retrieved in two parts and the last 19,825 characters were not read, so nothing is claimed about what is on the data-and-formulae sheet.
The International System of Units (SI), 9th edition 2019, version V4.01 of June 2026, Bureau International des Poids et Mesures, bipm.org/documents/d/guest/si-brochure-9-en-pdf (read 11 November 2026). Cited for the units, from Table 4 of coherent derived units with special names: magnetic flux, weber (Wb) = kg m² s⁻² A⁻¹ = V s; magnetic flux density, tesla (T) = kg s⁻² A⁻¹ = Wb/m²; inductance, henry (H) = kg m² s⁻² A⁻² = Wb/A. Those three equalities are worth more than they look: a weber is a volt-second, so a flux change of one weber in one second is one volt — which is Faraday’s law written as a unit. And a henry is a weber per ampere, so an inductance is just how much flux linkage one ampere produces.
University Physics Volume 2, chapters 11 and 14, OpenStax, Rice University (openstax.org/books/university-physics-volume-2, key-equations pages read 11 November 2026). An openly licensed university text, cited here because the coaching sites that dominate searches for this topic are not citable and mostly copy one another. Chapter 11 gives the radius of a charged particle’s path as r = mv/(qB), its period as T = 2πm/(qB), the Hall potential as V = IBl/(neA), the drift velocity in crossed fields as v_d = E/B, the magnetic dipole moment as μ = NIA and the torque as τ = μ × B. Chapter 14 gives L_solenoid = μ₀N²A/l, NΦ = LI, ε = −L dI/dt, M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂ and U = ½LI². Neither key-equations page lists a cyclotron frequency or a field energy density, so those two are stated here from the definitions and not attributed to this source.