Electromagnetic Induction & Alternating Current

Changing flux, induced emf, inductance and AC circuits · Class 12 Physics

1The big idea

The previous chapter said a current makes a magnetic field. This one says the reverse — a changing magnetic field makes a current — and that single sentence is what powers your house. It is the principle behind generators, transformers, induction cooktops and the charger on your phone. The whole chapter grows from one law: whenever the magnetic flux threading a loop changes, an electromotive force (emf) appears around that loop to oppose the change.

The word to keep hold of is changing. A loop sitting in a strong but steady field has no induced emf at all. Move it, rotate it, shrink it, or vary the field — do anything that alters the flux through it — and only then does a current flow. Alternating current (AC), the second half of the chapter, is what you get when that change is made to repeat smoothly and endlessly, sixty or fifty times a second.

2Magnetic flux — counting the field lines

Before we can talk about a change in flux we need flux itself. Magnetic flux Φ measures how much magnetic field pokes through a surface — loosely, the number of field lines threading the loop. For a flat area A in a uniform field B:

Here θ is the angle between the field and the normal (the perpendicular) to the loop, not the loop's face. When the field is straight through the loop, θ = 0, cos θ = 1 and the flux is greatest, Φ = BA. When the field skims along the plane of the loop, θ = 90° and Φ = 0 — no lines pass through. The unit of flux is the weber (Wb), and one weber per square metre is one tesla.

Notice the three separate handles on Φ: the field B, the area A, and the orientation θ. Change any of them and you change the flux — which, as the next section shows, is how every generator and motor is built.

3Faraday's law & Lenz's law

Faraday's experiments boil down to one quantitative statement: the induced emf equals the rate at which the flux is changing. If the loop is a coil of N turns, each turn feels it, so they add:

Read it slowly. The emf depends not on the flux but on its rate of change. A slow change gives a feeble emf; a sudden one gives a large emf. Swipe a magnet quickly past a coil and the needle kicks hard; ease it past and the needle barely stirs.

What is that minus sign doing? That is Lenz's law, and it is not a bookkeeping nuisance — it is energy conservation wearing a disguise. Lenz's law says: the induced current always flows in the direction that opposes the change producing it. Push a magnet's north pole toward a coil and the coil's near face becomes a north pole to push back; pull it away and the near face turns south to pull it back.

4Motional emf — deriving BLv from scratch

Here is the cleanest place to see Faraday's law give a formula. Slide a straight conducting rod of length L along two parallel rails at speed v, with a uniform field B pointing straight out of the plane. The rod, rails and a resistor close the loop.

Let the rod sit a distance x from the closed end. The area of the loop is A = L x, so the flux (field straight through, cos θ = 1) is Φ = B L x. Now the rod moves, so x grows at rate dx/dt = v. Differentiate:

That is motional emf: a rod of length L cutting field lines at speed v develops an emf B L v across its ends. You can reach the same answer from the magnetic force on the free charges in the rod (each feels q v B, which pushes them to one end until an electric field balances it), and the two pictures agree exactly. This is the heart of every generator: spin a coil in a field and each side is a rod of the sort above, sweeping out an emf that reverses smoothly twice per turn — an alternating emf.

5Self- and mutual inductance

A coil carrying a current makes its own flux, and that flux threads the coil itself. If you change the current, its own flux changes, so by Faraday's law the coil induces an emf in itself that opposes the change. This self-opposition is called self-inductance, L, defined by Φ = L I, giving the back-emf

Inductance is electrical inertia: it resists changes in current the way mass resists changes in motion. Try to switch the current on suddenly and the coil fights back; that is why a large inductor sparks when you break its circuit. Its unit is the henry (H).

Energy stored in an inductor

To build the current up from zero you must work against that back-emf. The power you supply at any instant is P = ε I = L I dIdt. Integrate from I = 0 to the final current I:

So an inductor carrying current I stores energy ½ L I², held in its magnetic field — the exact partner of ½ C V² for a capacitor's electric field. Kill the current and that energy has to go somewhere, which is why an inductor is so reluctant to let go.

Mutual inductance. Put a second coil nearby. A changing current in the first sends changing flux through the second and induces an emf there — with no wires between them. This is mutual inductance M, with ε₂ = −M dI₁dt. It is completely symmetric (the same M works both ways) and it is precisely how a transformer moves energy from its primary to its secondary across an air (or iron) gap.

6Alternating current & rms values

Spin a coil steadily in a magnetic field and the emf you get varies as a sine: V = V₀ sin ωt, driving a current I = I₀ sin ωt. Here V₀ and I₀ are the peak values and ω = 2πf is the angular frequency (India's mains runs at f = 50 Hz).

But if the current sloshes back and forth, averaging to zero over a cycle, what does it mean to say the mains is "230 volts"? The average is useless — power is what matters, and power depends on I², which is always positive. So we quote the root-mean-square (rms) value: square the current, average over a cycle, take the square root. Because the average of sin²ωt over a full cycle is exactly ½:

The same argument gives V_rms = V₀/√2. The rms value is the DC current that would deliver the same average heating power — that is its whole point. So "230 V mains" is the rms figure; its peak is V₀ = 230 × √2 ≈ 325 V, which is why insulation must be rated well above 230.

7Reactance, impedance & phase

Resistors, inductors and capacitors each respond differently to AC. A resistor obeys Ohm's law at every instant. An inductor and a capacitor also limit current, but their opposition — called reactance — depends on frequency, and crucially they shift the current out of step with the voltage.

• Inductive reactance X_L = ωL. An inductor resists changes, so the faster the AC (higher ω), the harder it fights: reactance rises with frequency. The current lags the voltage by 90°.
• Capacitive reactance X_C = 1ωC. A capacitor passes rapid changes easily, so its reactance falls as frequency rises. The current leads the voltage by 90°.

Put R, L and C in series and their oppositions do not simply add, because they are 90° apart in timing. You combine them like perpendicular vectors (phasors): the resistance along one axis, the net reactance X_L − X_C along the other. The hypotenuse is the total opposition, the impedance Z:

and the current amplitude is just I₀ = V₀/Z, an Ohm's law for AC. The angle of that phasor triangle is the phase difference between voltage and current, tan φ = X_L − X_CR. When X_L > X_C the circuit is inductive and current lags; when X_C > X_L it is capacitive and current leads.

8Resonance & power factor

The activity shows the star result. As you raise the frequency, X_L = ωL climbs and X_C = 1/ωC falls. At one special frequency they are exactly equal, the net reactance vanishes, and the impedance collapses to just Z = R — its smallest possible value. The current surges to its maximum. Setting X_L = X_C:

This is series resonance, and it is how a radio tunes: adjusting C moves f₀ until it matches one station's frequency, where that station alone drives a large current and the rest are ignored.

Power factor and wattless current

In AC the average power is not simply V_rms I_rms. Because current and voltage are out of step by φ, only the in-phase part of the current does work:

The factor cos φ = R/Z is the power factor. A pure resistor has φ = 0, cos φ = 1, full power. A pure inductor or capacitor has φ = 90°, cos φ = 0 — it takes current but consumes no average power at all. That current, sloshing energy in and out of the field twice per cycle without net loss, is the wattless current. At resonance the reactances cancel, φ = 0, and the power factor is a perfect 1.

9The transformer

A transformer is mutual inductance put to work. Two coils share an iron core: the primary (N_p turns) carries the incoming AC, whose changing flux runs through the core and threads the secondary (N_s turns). The same changing flux Φ passes through every turn, so each turn gets the same induced emf dΦ/dt — and the voltages are simply in the ratio of the turns:

More secondary turns than primary steps the voltage up; fewer steps it down. But a transformer creates no energy — for an ideal one, power in equals power out, V_pI_p = V_sI_s. So whatever you gain in voltage you pay for in current: a step-up transformer raises voltage but lowers current in the same proportion. That trade is why power is sent across the country at hundreds of thousands of volts — high voltage means low current, and low current means small I²R heating losses in the wires — then stepped back down to a safe 230 V near your home. A transformer works only on AC, precisely because it needs a changing flux; feed it steady DC and the secondary is dead.

10Common confusions to clear up

• Flux, not field, drives the emf. A loop can sit in an enormous steady field and generate nothing. Only a changing flux — from moving, rotating or a varying field — induces an emf.
• Lenz's law opposes the change, not the current. The induced current opposes the change in flux that made it, which is why it fights you whether you push the magnet in or pull it out.
• rms is not the average. The plain average of a sine over a cycle is zero. The rms value is the square-root of the mean square, equal to peak divided by √2, and it is what "230 V" means.
• Reactance is not resistance. A resistor dissipates energy; a pure inductor or capacitor stores and returns it, consuming zero average power despite carrying current — the wattless current.
• At resonance the impedance is minimum, not maximum, for a series LRC circuit. The reactances cancel, so Z = R and the current peaks.
• A transformer changes voltage, never frequency, and needs AC to function. It cannot step up power, only trade voltage for current.

11Check yourself

Class 12 Physics · Electromagnetic Induction & Alternating Current · aligned to NCERT Chapters 6–7 · SmartStudy.School