Physics · Electromagnetism

Faraday's Law and Lenz's Law

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What you learned

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Key takeaway

Induced emf depends on changing flux linkage $λ = N Φ$ , not on magnetic field strength alone.
A coil can link strong magnetic flux and still have zero induced emf if the flux linkage is not changing at that instant.

Common misconception

If the magnetic field through the coil is large, the induced emf must also be large.

A large field can give a large flux linkage, but induction depends on the rate of change of flux linkage. A flat flux curve gives zero emf even when the field is strong.

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Equation snapshotOne-dimensional magnet pass · Bounded field-through-coil modelShow 2 equations

One-dimensional magnet pass
$x_{m} (t) = x_{0} + v t$
The magnet's position along the shared axis is what changes the field through the coil during the pass.
Bounded field-through-coil model
$B_{coil} (x_{m}) = s \frac{B _{0}}{1 + ( x _{m} / ℓ ) ^{2}}$
In this compact model, the field through the coil is strongest near the coil center and weakens with distance. The pole sign $s = \pm 1$ flips the field direction.

Why it behaves this way

Explanation

Electromagnetic induction is about change. A magnetic field through a loop is not enough by itself. The coil develops an emf only when the magnetic flux linkage through it changes, and Lenz's law fixes the sign so the induced response opposes the change that produced it.

This module keeps one moving-magnet-and-coil picture in charge. One bar magnet passes one coil on one shared axis. The same magnet position, speed, pole orientation, coil turns, and coil area determine the stage, the galvanometer, the current arrows, the flux graph, and the induced-response graph, so Faraday's law and Lenz's law stay tied to one changing setup.

Key ideas

01Induced emf depends on changing flux linkage

λ = N Φ

, not on magnetic field strength alone.

02A coil can link strong magnetic flux and still have zero induced emf if the flux linkage is not changing at that instant.

03More turns, larger coil area, stronger magnetic coupling, or faster relative motion can increase the induction signal because they increase the flux linkage or how quickly it changes.

04The sign of emf and current is a direction label: positive and negative mean opposite loop directions in the stage convention, and Lenz's law chooses the direction that opposes the current flux change.

05Reversing the motion direction or flipping which pole faces the coil reverses the sign of the induced emf and current, because Lenz's law makes the response oppose the change in flux.

Worked examples

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Frozen walkthrough

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Frozen walkthrough

Use the live magnet pass directly. The same magnet position, speed, pole orientation, coil turns, and coil area set the stage, the galvanometer, the graphs, and the calculations below.

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Example 1 of 2

Frozen valuesUsing frozen parameters

At this moment in the magnet pass, what are the coil's flux linkage, induced emf, and current?

B_{0}

Magnet strength

1.4 T

N

Coil turns

120 turns

A

Coil area

1 m²

v

Magnet speed

1.2 m/s

x_{m}

Magnet position

-2.6 m

1. Read the magnet's current position and motion

The magnet is at

x_{m} = - 2.6 m

with

v = 1.2 m/s

, so it is approaching the coil.

2. Find the signed field through the coil

With pole sign

s = + 1

, the bounded field model gives

B_{coil} = 0.24 T

3. Calculate the linked flux

Using

N = 120

and

A = 1 m^{2}

, the coil links

Λ = N A B_{coil} = 0.52 Wb \cdot turn

in this bounded setup.

4. Use Faraday's law to find emf and current

Faraday's law gives

E = - d Λ/ d t = - 0.4 V

, so the loop current is

I = - 0.17 A

Flux linkage, emf, and current

Λ = 0.52 Wb \cdot turn, E = - 0.4 V, I = - 0.17 A

The linked flux is changing enough to drive a clockwise current in the stage convention, which is the model's Lenz-law response to the present change.

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The simulation shows one circular coil fixed at the center of the stage and one bar magnet sliding horizontally past it. A field band marks the signed magnetic field through the coil, a galvanometer card reports induced emf and current, and optional arrows show the loop-current direction when the response is not zero.

A live readout lists time, magnet position, field through the coil, flux-change rate, induced emf, and current. The same shared pass also drives the graphs, so hovering the time-based plots previews the corresponding moment on the stage.

Graph summary

The first graph compares the field through the coil with the linked flux. The flux curve has the same basic shape as the field curve, but it is scaled by coil turns and area.

The second graph compares induced emf with loop current over the same time axis. The key point is that this response follows how quickly the flux linkage changes, not whether the field or flux is merely large at one instant.

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Magnetism

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