Faraday's law

Hold a bar magnet still near a loop of wire and the loop's galvanometer needle does not so much as twitch — even though the magnet's field passes straight through the loop. Now push the magnet toward the loop and the needle jumps; pull it away and the needle jumps the other way, just as fast. Nothing touched the wire. No battery is connected. Yet a current flows, however briefly, purely because the magnetic field threading the loop was changing. Michael Faraday saw this in 1831 and spent months turning coils, sliding magnets and switching currents on and off nearby wires, cataloguing every arrangement that made a galvanometer needle move. Joseph Henry in America independently found the same effect. What both had stumbled onto is one of the deepest ideas in physics: a changing magnetic field creates an electric field, and that electric field can drive a current — the same fact, at bottom, that makes AC generators, wireless phone chargers, and card readers all work. Faraday's genius was to go beyond "it happens" to "here is exactly how much." He and later workers pinned the induced EMF to a single measurable quantity: the rate at which magnetic flux — the amount of magnetic field cutting through the circuit — changes with time. Double the rate of change and the EMF doubles; use a coil with twice as many turns and the EMF doubles again. This article turns that discovery into the working formula every CBSE Class 12 student must state, derive for a simple case, and apply to numbers: $\varepsilon = -N\,d\phi_B/dt$ . We will build the physical picture, derive the motional-EMF special case from force on a charge, work a board-style and a competitive-style numerical, and expose the traps that quietly cost marks — the missing $N$ , average versus instantaneous EMF, and the false belief that a steady field near a stationary loop induces anything at all.

Key takeaways

Faraday's law: whenever the magnetic flux $\phi_B$ linked with a circuit changes with time, an EMF is induced in it, equal in magnitude to the rate of change of flux linkage: $\varepsilon = -N\,d\phi_B/dt$ .
Flux can change three ways — recall $\phi_B = BA\cos\theta$ — by changing $B$ (a magnet moving, a nearby current switching), changing $A$ (a sliding rod on rails), or changing $\theta$ (a coil rotating in a field, the heart of a generator).
The negative sign is Lenz's law compressed into the formula: the induced EMF always opposes the change of flux that produces it; magnitude calculations usually just use $|\varepsilon|$ .
For a coil closed through resistance $R$ , the induced current follows Ohm's law directly: $I = |\varepsilon|/R$ .
A rod of length $\ell$ sliding with speed $v$ across a field $B$ generates a motional EMF $\varepsilon = B\ell v$ — the same law seen from the force-on-a-charge point of view, not a separate rule.
No relative change, no EMF: a stationary loop in a steady uniform field, however strong, has zero induced EMF because $d\phi_B/dt = 0$ .
Key idea: an EMF is induced in a circuit exactly when, and only when, the linked magnetic flux is changing, with magnitude $N\,d\phi_B/dt$ — one law that covers a moving magnet, a sliding rod, and a rotating coil alike.

The core insight

Faraday's law states that a changing magnetic flux linked with a circuit induces an EMF equal to $N$ times the rate of change of flux, $\varepsilon = -N\,d\phi_B/dt$ , and this single rule governs every way of "cutting" field lines — a moving magnet, a sliding rod, or a rotating coil.

Where it fits in the syllabus

Faraday's law is the quantitative centrepiece of the CBSE Class 12 chapter "Electromagnetic Induction" (Unit VI), coming right after magnetic flux and immediately before Lenz's law and self/mutual inductance. The prescribed scope is the law itself — $\varepsilon = -N\,d\phi_B/dt$ — and its use to compute induced EMF and induced current; a simple rod-on-rails illustration of motional EMF is allowed, but a full Lorentz-force derivation of motional EMF and any eddy-current or generator-construction detail belong to the next chapter (Alternating Current), where the generator formula is explicitly listed. This article keeps the derivation to that permitted simple case and stays entirely within board-scope numbers.

The physical picture

Magnetic flux through a flat surface of area $A$ , in a uniform field $B$ making angle $\theta$ with the surface's normal, was defined in the previous topic as $\phi_B = BA\cos\theta$ ; it is measured in weber (Wb), where $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$ . Faraday's experiments showed that it is not the flux itself but its rate of change that matters for inducing an EMF. For a coil of $N$ identical turns, each linked with the same flux $\phi_B$ , the total flux linkage is $N\phi_B$ , and the induced EMF is:

\varepsilon = -N\frac{d\phi_B}{dt}

where: $\varepsilon$ is the induced EMF in volts, $N$ is the number of turns in the coil, $\phi_B$ is the magnetic flux linked with one turn in weber, and $t$ is time in seconds. The physical reading is that $\varepsilon$ is literally a slope — the steepness with which flux is rising or falling at that instant, scaled up by however many turns share that same changing flux. If flux changes at a steady rate over a finite interval $\Delta t$ , the average induced EMF over that interval is simply:

\varepsilon = -N\frac{\Delta\phi_B}{\Delta t}

where: $\Delta\phi_B = \phi_{B,\text{final}} - \phi_{B,\text{initial}}$ is the net change in flux per turn over the interval $\Delta t$ . This average form and the instantaneous derivative form agree only when the flux changes uniformly (a straight-line graph of $\phi_B$ against $t$ ); if the loop rotates, for instance, flux varies sinusoidally and the two differ except at special instants. Once the circuit is closed through a total resistance $R$ , the induced current is fixed by Ohm's law applied to this induced EMF as the source:

I = \frac{|\varepsilon|}{R}

where: $I$ is the induced current in amperes and $R$ is the total resistance of the closed loop in ohms; the sign of $\varepsilon$ (from Lenz's law, taken up next) fixes the current's direction, but for a magnitude calculation $|\varepsilon|$ is used directly.

Figure

A bar magnet approaching a circular coil connected to a galvanometer, with the coil field lines shown passing through the loop and the galvanometer needle deflected, next to a second panel showing the magnet stationary and the needle at rest

The physical picture continued: three ways flux can change

Because $\phi_B = BA\cos\theta$ is a product of three quantities, there are exactly three independent ways to make it change, and Faraday's law treats all three identically — the formula only ever asks for $d\phi_B/dt$ , not why the flux is changing. First, $B$ itself can vary in time, as when a magnet approaches or recedes from a coil, or a neighbouring coil's current is switched on, off, or varied. Second, the area $A$ enclosed by the circuit can change, as when a conducting rod slides along a pair of rails, continuously altering the loop's enclosed area within a fixed field — this is the "motional EMF" case examined in the derivation below. Third, the orientation angle $\theta$ between the field and the loop's normal can change, as when a coil is mechanically rotated inside a uniform field; this rotation is the working principle of an AC generator, whose construction is developed fully in the next chapter, but the flux-change reasoning here already explains why rotating a coil produces an EMF at all.

Derivation AMotional EMF for a rod sliding on rails, from the force on a moving charge

The goal of this derivation is to show that Faraday's flux-rate law, $\varepsilon = -N\,d\phi_B/dt$ , is not an independent postulate when the flux changes because of a moving conductor — it can be obtained directly from the ordinary force a magnetic field exerts on a moving charge, giving the same answer by two different routes and building genuine confidence in the law. Consider a straight conducting rod of length $\ell$ resting on two parallel, stationary conducting rails a distance $\ell$ apart, the whole arrangement sitting in a uniform magnetic field $B$ pointing into the page and perpendicular to the plane of the rails. The rod is pulled along the rails with constant speed $v$ , so the area of the closed loop formed by the rod, the rails, and a fixed connecting end steadily grows or shrinks.

Step 1. Track the free charges inside the moving rod. Every free charge $q$ in the rod moves with the rod itself at velocity $v$ (in addition to any motion along the rod), and it sits in the field $B$ . The magnetic force on it is $\vec F = q\vec v \times \vec B$ , whose magnitude works out to $F = qvB$ and whose direction, by the right-hand rule with $v$ along the rails and $B$ into the page, points along the length of the rod itself. This is precisely what a battery's internal force does — it pushes charge along a specific direction inside a conductor.

Step 2. Convert that force into an effective field and then an EMF. A force per unit charge is exactly what an electric field represents, so the free charges experience an effective field of magnitude $F/q = vB$ directed along the rod. The EMF driving current around the loop is this effective field integrated along the length over which it acts, and since it is uniform along the rod's length $\ell$ : $\varepsilon = (vB)\times \ell = B\ell v$ .

Step 3. Recover the same result from the flux-rate law, to confirm the two pictures agree. As the rod slides a small distance $dx$ in time $dt$ , the enclosed area of the loop changes by $dA = \ell\,dx$ , so the flux changes by $d\phi_B = B\,dA = B\ell\,dx$ . Dividing by $dt$ and using $dx/dt = v$ : $\dfrac{d\phi_B}{dt} = B\ell\dfrac{dx}{dt} = B\ell v$ . Since the loop here has $N=1$ turn, Faraday's law gives $|\varepsilon| = d\phi_B/dt = B\ell v$ — identical to Step 2.

Step 4. State the result and its reach. The motional EMF of a rod of length $\ell$ moving with speed $v$ perpendicular to itself and to a field $B$ is $\varepsilon = B\ell v$ . This is not a separate law from Faraday's — it is Faraday's law specialised to the case where flux changes because the enclosed area is changing at constant $B$ , and the force-on-a-charge argument shows why the free electrons in the rod actually redistribute to build up that EMF, giving physical insight beyond the flux bookkeeping alone.

Worked examples

Board level - coil in a rising uniform field

A coil of $200$ turns and area $0.02\ \text{m}^2$ is placed with its plane perpendicular to a magnetic field. The field increases uniformly from $0.1\ \text{T}$ to $0.5\ \text{T}$ in $0.4\ \text{s}$ . Find the magnitude of the induced EMF, and the induced current if the coil's resistance is $10\ \Omega$ .

Step 1. Since the plane is perpendicular to $B$ , the angle between $B$ and the area's normal is $\theta = 0°$ , so $\cos\theta = 1$ and $\phi_B = BA$ at every instant — there is no orientation complication here, only the magnitude of $B$ is changing.

Step 2. Compute the initial flux per turn: $\phi_1 = B_1 A = 0.1 \times 0.02 = 2\times10^{-3}\ \text{Wb}$ .

Step 3. Compute the final flux per turn: $\phi_2 = B_2 A = 0.5 \times 0.02 = 1\times10^{-2}\ \text{Wb}$ .

Step 4. Find the change in flux per turn: $\Delta\phi_B = \phi_2 - \phi_1 = 1\times10^{-2} - 2\times10^{-3} = 8\times10^{-3}\ \text{Wb}$ .

Step 5. Apply Faraday's law with $N = 200$ turns over $\Delta t = 0.4\ \text{s}$ : $|\varepsilon| = N\dfrac{\Delta\phi_B}{\Delta t} = 200 \times \dfrac{8\times10^{-3}}{0.4} = 200 \times 0.02 = 4\ \text{V}$ .

Step 6. Find the induced current using Ohm's law with $R = 10\ \Omega$ : $I = \dfrac{|\varepsilon|}{R} = \dfrac{4}{10} = 0.4\ \text{A}$ .

The induced EMF is 4 V and the induced current is 0.4 A.

Competitive level - rod on rails compared with a shrinking loop

A square loop of side $10\ \text{cm}$ sits perpendicular to a uniform field $B = 0.6\ \text{T}$ and is pulled fully out of the field in $0.2\ \text{s}$ , its enclosed area shrinking uniformly from full area to zero. (a) Find the average induced EMF using the flux method. (b) If instead one side of the loop is a rod of length $10\ \text{cm}$ pulled out at constant speed, find that speed and confirm $\varepsilon = B\ell v$ gives the same EMF.

Step 1. Find the loop's area: $A = (0.1)^2 = 0.01\ \text{m}^2$ , and the initial flux (single turn, $N=1$ ): $\phi_1 = BA = 0.6 \times 0.01 = 6\times10^{-3}\ \text{Wb}$ ; the final flux, once the loop is fully outside the field, is $\phi_2 = 0$ .

Step 2. Apply the average-EMF form of Faraday's law over $\Delta t = 0.2\ \text{s}$ : $|\varepsilon| = \dfrac{|\Delta\phi_B|}{\Delta t} = \dfrac{6\times10^{-3}}{0.2} = 3\times10^{-2}\ \text{V} = 30\ \text{mV}$ .

Step 3. Recast the same motion as a rod of length $\ell = 0.1\ \text{m}$ sweeping out that same area in the same time: the rod effectively travels a distance equal to the loop's own side ( $0.1\ \text{m}$ , the extent by which the enclosed area shrinks along the direction of pull) in $0.2\ \text{s}$ , so its speed is $v = \dfrac{0.1}{0.2} = 0.5\ \text{m/s}$ .

Step 4. Apply the motional-EMF formula derived above: $\varepsilon = B\ell v = 0.6 \times 0.1 \times 0.5 = 0.03\ \text{V} = 30\ \text{mV}$ — exactly matching Step 2, confirming that the flux-rate method and the motional-EMF formula are the same physics viewed two ways.

The average induced EMF is 30 mV either way, corresponding to an effective rod speed of 0.5 m/s. Timing note: in a competitive exam, recognising that a "shrinking loop" problem and a "rod on rails" problem are the same calculation saves a full derivation — go straight to whichever of $N\,\Delta\phi_B/\Delta t$ or $B\ell v$ has the numbers already in a convenient form, and use the other only to sanity-check the answer as done here.

Common traps

Forgetting the number of turns $N$ in the coil formula, and quietly computing the EMF for just one loop's worth of flux change when the coil has many turns.
Believing Faraday's law only applies when $B$ itself changes — flux can equally change through a changing area (rod on rails) or a changing orientation (rotating coil), and the same $\varepsilon = -N\,d\phi_B/dt$ governs all three.
Mixing up average EMF, $\Delta\phi_B/\Delta t$ over a finite interval, with instantaneous EMF, the derivative $d\phi_B/dt$ at one instant — these coincide only when flux changes at a constant rate, not when it varies sinusoidally, as in a rotating coil.
Assuming that a strong, perfectly steady magnetic field threading a stationary loop must induce something — it induces nothing at all, because $d\phi_B/dt = 0$ regardless of how large $B$ is.

You should be able to…

State Faraday's law of electromagnetic induction in words and as the formula $\varepsilon = -N\,d\phi_B/dt$ , and explain what the minus sign represents.
Identify, for a given physical situation, which of $B$ , $A$ , or $\theta$ is changing to produce the induced EMF.
Calculate average induced EMF from a given initial and final flux and a time interval, correctly including the number of turns $N$ .
Derive the motional EMF $\varepsilon = B\ell v$ for a rod sliding on rails from the force on a moving charge, and show it agrees with the flux-rate calculation.
Calculate induced current from induced EMF and circuit resistance using Ohm's law, $I = |\varepsilon|/R$ .
Explain why a steady field through a stationary loop induces no EMF, however strong the field is.

Ask the tutor: Want to practice spotting which of $B$ , $A$ , or $\theta$ is changing in a tricky diagram, or work through more rod-on-rails and rotating-coil EMF problems together?

Why is there a minus sign in Faraday's law if we only ever seem to use the magnitude?

The minus sign is not decorative — it is Lenz's law, the statement that the induced EMF always opposes the change in flux that creates it, built directly into the formula. In most numerical problems you are only asked for the size of the induced EMF or current, so you compute $|\varepsilon| = N\,|d\phi_B/dt|$ and drop the sign. But whenever a question asks for the direction of the induced current, or why energy is conserved during induction, the minus sign is exactly where that physics lives, and you switch to reasoning with Lenz's law directly (the next topic in this chapter) rather than chasing the sign through the formula algebraically.

If I spin a coil twice as fast in the same field, does the induced EMF just double?

For a coil rotating in a uniform field, the flux varies as $\phi_B = BA\cos(\omega t)$ , so $d\phi_B/dt = -BA\omega\sin(\omega t)$ , and the peak EMF is proportional to $\omega$ , the angular speed — so yes, doubling the rotation rate doubles the peak induced EMF. This case is the working principle of an AC generator, whose full formula $\varepsilon = NBA\omega\sin(\omega t)$ and construction are developed in the next chapter on Alternating Current; for this chapter, the important point is simply that a faster rate of flux change from rotation, exactly like a faster rate of change from a moving magnet, produces a larger instantaneous EMF, consistent with Faraday's law.

Does the wire have to be a closed loop for an EMF to be induced?

No. A changing magnetic flux induces an EMF along any conducting path linking that flux, whether or not the circuit is closed — Faraday actually discovered the effect using open-ended arrangements before settling on closed loops for easy current measurement. What a closed loop and a finite resistance give you is a current you can detect, via $I = |\varepsilon|/R$ ; an open circuit still has an EMF appearing between its ends (you could measure it with a voltmeter, which draws negligible current), it simply has no complete path for charge to keep flowing around. The "closed loop" language in the syllabus formula is there because CBSE numericals almost always ask for current as well as EMF, not because a closed loop is a strict requirement for the EMF to exist.

How is this different from the electric field around a stationary charge that we studied in electrostatics?

They are genuinely different kinds of electric field. The electrostatic field from a charge is conservative — it can be derived from a potential, and it does zero net work carrying a charge around any closed loop. The electric field induced by a changing magnetic flux is fundamentally non-conservative: it does net work on a charge taken around a closed loop, which is exactly why it can drive a continuous current in a coil (in electrostatics, a static field could never sustain a current around a closed wire, since that would violate energy conservation). This distinction — a curling, non-conservative induced electric field versus the radial, conservative electrostatic field — is one reason electromagnetic induction was such a conceptually new discovery in 1831, not just a variation on Coulomb's law.