MHT-CET Physics · Electromagnetic Induction

Faraday's Law, Lenz's Law and Induced Charge

Magnetic flux is B·A; whenever the flux through a circuit changes, an e.m.f. equal to the rate of change of flux linkage is induced (Faraday), in the direction that opposes the change (Lenz), and the charge that flows depends only on the total change in flux and the resistance, not on how fast it happened.

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Why this matters

22 PYQs, none HARD. Fifteen are Faraday's law with numbers — flux given as a function of time, a coil pulled out of a field, a field cut to a quarter, the charge that flows; seven are Lenz's law — a magnet falling through a ring, a cut ring or a pipe, the direction of the current in a ring falling towards a wire, and why Lenz's law is energy conservation. Two cards.

Concept 1 of 2: Flux, Faraday's Law and Induced Charge

Flux counts the field lines through an area: φ = B·A = BA cos θ, so a field component lying IN the plane of a coil passes through nothing. Faraday's law says the e.m.f. equals the rate of change of flux linkage, e = N dφ/dt. If φ is given as a function of time, differentiate; if a coil is simply pulled out of a field in time t, the average e.m.f. is NBA/t. The current is e/R. And the CHARGE that flows is current × time = NΔφ/R — the time cancels, so a fast or a slow pull moves the same charge.

Definition

Flux $\phi = \vec B \cdot \vec A$ : a square of side L in the x–y plane in $\vec B = B_0(2\hat i + 3\hat j + 4\hat k)$ links only the $\hat k$ part, $4B_0L^2$ .
Faraday: $e = -N\dfrac{d\phi}{dt}$ ; average $e = \dfrac{N\,\Delta\phi}{t}$ . $\phi = 50t^2 + 4$ ⇒ $e = 100t$ .
Current $I = \dfrac{e}{R_{\text{total}}}$ (include any series resistance: coil R plus R/2 gives 3R/2).
Charge $q = \dfrac{N\,\Delta\phi}{R}$ : depends on the total change of flux, not its rate.
Field falling to 25% of B in time t: $e = \dfrac{3BA}{4t}$ .

Faraday's law and induced charge

e = -N\frac{d\phi}{dt}, \qquad q = \frac{N\,\Delta\phi}{R}

Worked example

A 40-turn coil of area 50 cm² lies normal to a 0.05 T field and is pulled out in 0.02 s. Coil resistance 5 Ω. Average e.m.f. and charge that flows?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 22 April Shift I · Q50Easy

Example 1 · Electromagnetic Induction · Faraday's and Lenz's Laws — Induced EMF, Current, and Charge

A coil of resistance

400\Omega

is placed in a magnetic field. If the magnetic flux '

\phi

, (Wb) linked with the coil varies with time '

t

' (

s

) as

\phi= 50t^{2}+ 4

, the current in the coil at

t = 2\text{ }s

will be

Thinking a faster change moves more charge

A faster change gives a bigger e.m.f. and current, but for less time. The charge q = NΔφ/R is the same however quickly the flux changes.

Using the whole field for the flux

Only the component of B perpendicular to the coil passes through it. For a square in the x–y plane,

B = B_0(2\hat{i} + 3\hat{j} + 4\hat{k})

gives 4B₀L², not √29 B₀L².

Drill 14 more on flux, faraday's law and induced charge

Concept 2 of 2: Lenz's Law: the Induced Current Opposes the Change

The induced current always pushes back against whatever changes the flux. A magnet falling through a closed ring is braked, so it falls with less than g; a hollow metal pipe does the same through eddy currents. A ring with a CUT carries no current, so the magnet falls freely at g. A ring falling towards a wire whose field points out of the page above it sees that outward flux grow, so its current makes an inward field — clockwise. And if the coil and magnet move together, nothing changes and nothing is induced. This opposition is why Lenz's law is the law of conservation of energy in disguise: you must do work to push against it.

Definition

Closed ring or metal pipe, magnet dropped through: acceleration less than g. Cut ring: exactly g (e.m.f. but no current).
Coil and magnet moving together: no relative motion, zero e.m.f.
Ring falling towards a straight current (field out of the page above the wire): induced current clockwise.
North pole moving away from a loop: the induced current tries to keep the flux, attracting the magnet back.
Lenz's law is a statement of conservation of energy.

Situation	Induced effect
Magnet dropped through a closed ring or pipe	falls with acceleration less than g
Magnet dropped through a cut ring	falls with g — no current An e.m.f. is still induced across the cut.
Coil and magnet moving together	no e.m.f.
Flux into a loop increasing	current makes a field out of it
Flux into a loop decreasing	current makes a field into it

The induced current opposes the CHANGE in flux, not the flux itself.

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 21 April Shift II · Q27Moderate

Example 2 · Electromagnetic Induction · Faraday's and Lenz's Laws — Induced EMF, Current, and Charge

A conducting ring of certain resistance is falling towards a current carrying straight long conductor. The ring and conductor are in the same plane. Then

Opposing the flux instead of its change

When the flux is DECREASING, the induced current tries to keep it — its field points the SAME way as the original. It opposes the change, not the field.

Drill 6 more on lenz's law: the induced current opposes the change

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Flux, Faraday's Law and Induced Charge
Faraday's law and induced charge
$e = -N\frac{d\phi}{dt}, \qquad q = \frac{N\,\Delta\phi}{R}$

Reference tables (1)

Lenz's Law: the Induced Current Opposes the Change5 rows

Situation	Induced effect
Magnet dropped through a closed ring or pipe	falls with acceleration less than g
Magnet dropped through a cut ring	falls with g — no current An e.m.f. is still induced across the cut.
Coil and magnet moving together	no e.m.f.
Flux into a loop increasing	current makes a field out of it
Flux into a loop decreasing	current makes a field into it

The induced current opposes the CHANGE in flux, not the flux itself.

Watch out for (3)

Thinking a faster change moves more charge
→ Flux, Faraday's Law and Induced Charge
Using the whole field for the flux
→ Flux, Faraday's Law and Induced Charge
Opposing the flux instead of its change
→ Lenz's Law: the Induced Current Opposes the Change

Test yourself on Electromagnetic Induction

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