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Concept module

Faraday's Law and Lenz's Law

Track one magnet passing one coil and see how changing magnetic flux linkage creates induced emf while Lenz's law fixes the response direction, with the stage, galvanometer, and graphs all driven by the same bounded motion.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 4.50 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s4.50 s

Faraday's Law and Lenz's Law

One coil stays fixed while one bar magnet passes across the same axis. The field through the coil, linked flux, induced emf, galvanometer deflection, and current arrows all come from the same bounded moving-magnet state.

Graphs

Switch graph views without breaking the live stage and time link.

Field and flux linkage

The field-through-coil trace and the linked-flux trace come from the same magnet pass, so the taller flux curve is just the field story scaled by coil turns and area.

time (s): 0 to 4.5signed magnetic linkage response: -4 to 4

Field through coilFlux linkage

Hover or scrub to link the graph back to the stage.time (s) / signed magnetic linkage response

Controls

Adjust the physical parameters and watch the motion respond.

Magnet strength

B_{0}

1.4 T

Changes the field-through-coil scale in the bounded magnet model.

Coil turns

N

120 turns

Adds or removes turns from the same loop, which changes the flux linkage without changing the basic pass.

Coil area

A

1 m^2

Changes the size of the loop that links the field through the coil.

Magnet speed

v

1.2 m/s

Positive values start on the left and move right. Negative values start on the right and move left.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Hide

Starting offset

x_{0}

2.6 m

Sets the initial magnet distance from the coil center at t = 0.

North pole faces coil

s

Flip which pole points toward the coil to reverse the sign of the induction story.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Try thisPrompt 1 of 1

Flip the pole or reverse the motion and the response sign flips. Lenz's-law direction is always tied to opposing the change in flux.

Try this

Keep the same pass and use the reverse-polarity preset. Watch the field band, meter sign, and current-loop arrows reverse together.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

B_{0}

Magnet strength

1.4 T

Changes the size of the magnetic coupling through the coil, which lifts both the flux linkage and the induction response.

Graph: Field and flux linkageGraph: Induced emf and currentOverlay: Field bandOverlay: Current loop

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the current prompt as a compact investigation cue. Each one points at an induction pattern the stage, galvanometer, and graphs already show in the live state.

Try thisPrompt 1 of 1

Graph: Field and flux linkage

Flip the pole or reverse the motion and the response sign flips. Lenz's-law direction is always tied to opposing the change in flux.

Try this

Keep the same pass and use the reverse-polarity preset. Watch the field band, meter sign, and current-loop arrows reverse together.

Why it matters

It ties sign conventions, direction, and opposition to change into one shared picture instead of three disconnected rules.

Control: North pole faces coilControl: Magnet speedGraph: Field and flux linkageGraph: Induced emf and currentOverlay: Field bandOverlay: Current loopEquation

s

Equation

v

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Field band

Shows the signed field threading the coil at the current moment.

What to notice

The band color flips when the pole facing the coil flips, even if the rest of the geometry stays the same.

Why it matters

It keeps the magnetic part of the induction story visible instead of hiding it behind the meter.

Control: Magnet strengthControl: North pole faces coilGraph: Field and flux linkageEquation

B_{0}

Equation

s

Challenge mode

Use the same bounded coil-and-magnet pass to hit one honest Faraday/Lenz target. The checklist reads the live flux and emf state instead of a detached answer key.

0/2 solved

ConditionStretch

3 of 8 checks

Oppose the rising flux

Pause during the left-side approach so the magnet is still outside the coil, the linked flux is increasing, and the induced current runs in the clockwise Lenz response.

Inspect timeGraph-linkedGuided start2 hints

Suggested start

Start from the default pass, pause on the left approach, and use the current-loop arrow together with the field and flux graph.

Matched

Open the Field and flux linkage graph.

Field and flux linkage

Matched

Keep the Field band visible.

Matched

Keep the Current loop visible.

Pending

Pause into inspect mode.

live

Pending

Inspect the left-approach window near t = 1.8 s.

0 s

Pending

Keep the magnet left of center, around x = -0.55 to -0.3 m.

-2.6 m

Pending

Keep the linked flux increasing above 1.6 Wb-turn/s.

0.4

Pending

Make the induced current stay in the clockwise band, about -0.95 to -0.65 A.

-0.17 A

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, a 1.4 T magnet with north faces coil sits at x = -2.6 m and moves left-to-right at 1.2 m/s. The coil links 0.52 Wb-turn of flux, so the induced emf is -0.4 V and the current is -0.17 A. Flux linkage is increasing, so the coil responds with an induced emf that opposes that increase. The loop current is clockwise in the stage convention.

Equation detailsDeeper interpretation, notes, and worked variable context.

One-dimensional magnet pass

The same shared pass determines where the magnet is relative to the coil at every instant.

x_{m} (t) = x_{0} + v t

Positive v means the magnet starts on the left and moves right.

Negative v means the magnet starts on the right and moves left.

v

Magnet speed 1.2 m/s

x_{0}

Starting offset 2.6 m

Bounded field-through-coil model

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.

B_{coil} (x_{m}) = s \frac{B _{0}}{1 + ( x _{m} / ℓ ) ^{2}}

B_{0}

Magnet strength 1.4 T

x_{0}

Starting offset 2.6 m

s

Pole sign On

Flux linkage

More turns or more coil area means more linked magnetic flux for the same field-through-coil snapshot.

Λ = N Φ = N A B_{coil}

B_{0}

Magnet strength 1.4 T

N

Coil turns 120 turns

A

Coil area 1 m^2

s

Pole sign On

Faraday's law

Induced emf depends on how quickly the linked flux changes, not on the flux value alone.

E = - \frac{d Λ}{d t}

B_{0}

Magnet strength 1.4 T

N

Coil turns 120 turns

A

Coil area 1 m^2

v

Magnet speed 1.2 m/s

s

Pole sign On

Loop current

With one fixed loop resistance in this model, the induced current follows the induced emf directly.

I = \frac{E}{R _{loop}}

N

Coil turns 120 turns

v

Magnet speed 1.2 m/s

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 2 of 30 / 3 complete

Magnetism

Earlier steps still set up Faraday's Law and Lenz's Law.

1. Magnetic Fields2. Faraday's Law and Lenz's Law3. Magnetic Force on Moving Charges and Currents

Previous step: Magnetic Fields.

Start from track beginning

Short explanation

What the system is doing

Faraday's law is the bridge moment where magnetism stops being just a field pattern and starts becoming a source of electrical response. A magnetic field through a loop is not enough by itself. The loop only develops an emf when the magnetic flux through that loop changes, and Lenz's law fixes the sign so the induced effect opposes the change that produced it.

This module keeps one compact 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, the induced-response graph, the worked examples, the prediction prompts, the checkpoint challenge, and the quick test, so the Faraday/Lenz story stays tied to one honest changing setup instead of turning into a detached formula rule.

Key ideas

01Induced emf depends on changing flux linkage $\Lambda = N\Phi$, not on the mere presence of a magnetic field.

02Relative change matters. A strong magnetic field can exist through the coil while the induced emf is still zero if the flux is not changing at that instant.

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

04Reversing the motion direction or flipping which pole faces the coil reverses the sign of the induced emf and current. Lenz's law says that sign always opposes the change in flux.

Live induction checks

Solve the exact state on screen.

Work directly from the live moving-magnet state on screen. The same magnet pass, coil geometry, and polarity now driving the stage also set the flux linkage, emf, and current below.

Live valuesFollowing current parameters

For the current magnet pass, what flux linkage, induced emf, and loop current does the coil have right now?

B_{0}

Magnet strength

1.4 T

N

Coil turns

120 turns

A

Coil area

1 m^2

v

Magnet speed

1.2 m/s

x_{m}

Magnet position

-2.6 m

1. Read the live magnet pass

The magnet is at

x_{m} = - 2.6 m

with

v = 1.2 m/s

, so it is approaching the coil.

2. Evaluate the signed field through the coil

With pole sign

s = + 1

, the bounded field model gives

B_{coil} = 0.24 T

3. Build 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. Turn changing flux into emf and current

Faraday's law gives

E = - d Λ/ d t = - 0.4 V

, so the loop current is

I = - 0.17 A

Induction state

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.

Flux-change checkpoint

Suppose the magnet is centered in the coil and the coil is linking a strong field at that instant. Should the galvanometer necessarily show a large deflection? Answer from flux-change logic, not from field-size intuition.

Prediction prompt

Predict what the induced-response graph should do at the instant the flux curve reaches a maximum or minimum.

Check your reasoning

No. The galvanometer can read zero even at a strong-field moment if the flux linkage is not changing right then.

Faraday's law cares about

d Λ/ d t

, not just

Λ

. At the top or bottom of the flux curve, the slope is zero, so the induced emf and current cross through zero even though the field and linked flux can still be large.

Common misconception

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

A large field can produce a large flux, but induction depends on the rate of change of flux. A flat flux curve gives zero emf even at a strong-field moment.

That is why the induced-response graph crosses zero when the magnet is centered in a symmetric pass: the flux is momentarily at an extremum, so its slope is zero there.

Quick test

Reasoning

Question 1 of 4

Answer from flux-change reasoning, not from isolated formulas or memorized slogans.

A coil sits in a steady magnetic field that is not changing through time. Which statement is best?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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 second graph compares induced emf with loop current over the same time axis.

The key accessibility takeaway is that the response graph depends on how quickly the linked flux changes, not on whether the field or flux itself is merely large at one instant.

Keep the electricity-magnetism story moving

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Tie the four laws togetherElectromagnetism

Faraday's Law and Lenz's Law

See each variable before you move it.

Field band

Oppose the rising flux

Magnetism

What the system is doing

Solve the exact state on screen.

For the current magnet pass, what flux linkage, induced emf, and loop current does the coil have right now?

A coil sits in a steady magnetic field that is not changing through time. Which statement is best?

Keep moving through the concept map.

Maxwell's Equations Synthesis

Electromagnetic Waves

Magnetic Force on Moving Charges and Currents