Chapter 05 · Electromagnetic Induction · Full book

Electromagnetic Induction

Core intuition: Chapter Roadmap

Every earlier chapter treated magnetism as a consequence of moving charge. This chapter runs the arrow backwards: a changing magnetic situation drives charge to move. We begin with the single quantity that makes “changing” precise — magnetic flux — and build Faraday's law (\(\mathcal{E}=-d\Phi_B/dt\)) as the chapter's spine. Lenz's law fixes the direction and ties the whole effect to energy conservation. From there we read the law in two directions: outward, to the EMFs of moving conductors (the sliding rail, the rotating rod, the ac generator); and inward, to a circuit's response to its own changing current: self- and mutual induction, the energy stored in a magnetic field, and eddy currents. We close with the two canonical time-dependent circuits that induction makes possible: the RL growth/decay and the LC oscillation, the electrical twin of simple harmonic motion.

Perplexing Questions

Perplexing questions

A bar magnet rests motionless inside a closed copper ring. No current flows in the ring. The moment you pull the magnet away, a current appears — then vanishes the instant the magnet stops again. What is the ring actually responding to?
Two coils face each other. When the current in the first coil is steady, the second coil does nothing. The moment you switch the first coil off, a current surges through the second — in the opposite direction to what you might expect. Why does switching off produce a response, but a steady current does not?
A copper ring falls through a region of strong magnetic field. It slows down — measurably — even though nothing is physically touching it. What force is opposing its fall, and where does it come from?
The same physical law that governs generators in power stations, the braking system in high-speed trains, and the transformer in your phone charger can be written in a single compact equation. What could possibly unify such different devices?
Can you create a current in a conductor without any battery, any chemical reaction, or any direct contact with another current-carrying wire? If so, what replaces the energy source?

Why Changing Magnetism Creates Electricity

The previous chapter completed one half of a striking symmetry. Electricity produces magnetism. Every current-carrying wire is surrounded by a magnetic field; every moving charge is a source of magnetic influence on its neighbourhood. The natural question — one that troubled Faraday for years — is whether the reverse is also true. Can magnetism, under the right circumstances, produce electricity?

The answer is yes, but with a critical qualifier that the nineteenth century took time to articulate: a steady magnetic field produces nothing. A strong magnet held stationary above a copper ring leaves that ring completely inert. No deflection of any connected galvanometer, no warmth in the wire, no sign of any electrical effect whatsoever. The field is there; the conductor is there; and yet nothing happens.

This is the central surprise, and it demands a careful physical account.

The Role of Change

Place a bar magnet near a coil connected to a sensitive galvanometer. Hold the magnet still: the needle rests at zero. Now move the magnet toward the coil: the needle deflects. Stop the magnet — even close to the coil — and the needle falls back to zero. Pull the magnet away: the needle deflects in the opposite direction, then returns to zero when the motion stops.

The coil is not responding to the magnet's presence. It is responding to the change in the magnetic situation at its location.

Core intuition: Resolution of Perplexing Question 1

This is exactly the ring-and-magnet puzzle that opened the chapter. The ring responds not to the magnet's presence but to the change in the magnetic flux threading it. While the magnet is pulled away the flux is changing, so a current flows; the instant the magnet stops, the flux is again constant and the current vanishes. The full quantitative account is Faraday's law (§); its direction is Lenz's law (§).

The same effect can be produced without moving anything at all. Replace the bar magnet with a second coil carrying a current. While that current is steady, the galvanometer reads zero. Switch the current on: the galvanometer kicks. Switch it off: it kicks in the opposite direction. Increase the current slowly: a steady small deflection appears. The conductor is responding to a changing magnetic field, regardless of whether the change is produced by motion or by altering the current in a nearby source.

Core intuition: The Leap from Magnetic Effects to Induction

In the chapter on Moving Charges and Magnetism we established that a current is a source of magnetic field. Faraday's insight — confirmed experimentally in 1831 — is the converse: a changing magnetic field is a source of electrical driving force. The word “changing” is not a detail. It is the entire content of the phenomenon. A static magnetic field, however strong, drives no current in a stationary conductor. Only when the magnetic situation at the conductor's location is in the process of changing does an induced electromotive force (emf) appear.

What Counts as a Changing Magnetic Situation?

Induction arises whenever the magnetic environment threading through a conducting loop is in the process of changing. This can happen in several physically distinct ways, all governed by the same underlying law:

The strength of the external magnetic field changes with time (a nearby electromagnet being switched on or ramped up).
The loop moves into or out of a region of field, so the field it intercepts changes even if the field itself is static.
The area of the loop changes (a deformable loop being stretched or compressed).
The orientation of the loop relative to the field changes (the loop rotates — the principle of every electrical generator).

These four situations look different mechanically, but they are all manifestations of the same physical process, and Faraday's law will unify them under a single statement.

Induced EMF and Induced Current

When the magnetic situation at a conducting loop is changing, a driving force appears around the loop. This driving force is called the induced electromotive force, or induced emf. If the loop forms a complete circuit, the induced emf drives a current through the conductor: this is the induced current.

The induced emf is not a charge stored anywhere, nor a property of the conductor's material — it is a consequence of the changing magnetic environment. If the loop is open (a break in the wire), an emf still exists across the gap; it simply cannot drive a sustained current.

Misconception alert: “A Strong Field Always Induces a Current”

“A strong magnetic field will always induce a current.” This is incorrect. Magnitude of the field is irrelevant on its own. A superconducting magnet with a field of 10 T, held absolutely stationary next to a copper ring, induces precisely zero current. Induction depends entirely on change in the magnetic situation at the loop's location — not on the field's strength, not on its direction, and not on the mere proximity of a magnet. Students who memorise “magnets cause currents” will fail questions where the magnet is present but stationary, or where the field is uniform and unchanging across the loop.

✓ Checkpoint · Why Changing Magnetism Creates Electricity

A circular coil is placed in a uniform magnetic field directed perpendicular to the plane of the coil. The field magnitude is suddenly doubled and then held constant at the new value. Is an induced current present (a) while the field is increasing? (b) after the field has stabilised at the new value? Explain.
(a) Yes — the field through the coil is changing, so an induced emf and current appear. (b) No — once the field is again constant, there is no change in the magnetic situation and the induced emf drops to zero.
Identify which of the following produce an induced emf in a fixed rectangular loop: (i) a steady current in a nearby wire; (ii) a current in a nearby wire that is being increased; (iii) a bar magnet moving parallel to, but entirely outside, the plane of the loop so that no field lines pass through the loop; (iv) the loop being rotated in a uniform magnetic field.
(ii) and (iv) produce an induced emf. (i): steady current means steady field — no change, no induction. (iii): if no flux passes through the loop, rotating or moving sources outside can only induce emf if flux through the loop changes; here the loop is in the field's fringe at most — the key is whether the field threading the loop changes, not whether the source moves. (iv): rotation changes the orientation angle and hence the effective flux through the loop.
A conducting loop and a long straight wire carrying a steady current lie in the same plane. The loop is moved along the wire (parallel to it) at constant velocity. Is an emf induced? Now the loop is moved perpendicular to the wire. Is an emf induced in this case?
Along the wire: by symmetry, the field through the loop does not change as it moves parallel to an infinite straight wire in the same plane — no emf. Perpendicular to the wire: the loop moves into regions of stronger or weaker field, so the flux through it changes — emf is induced.

Magnetic Flux

◇ Figure 1

Fig 1. Magnetic flux through a loop. Only the component of \(\vec{B}\) along the area normal \(\hat{n}\) threads the surface: \(\Phi_B = BA\cos\theta\), where \(\theta\) is the angle between \(\vec{B}\) and \(\hat{n}\).

The previous section established that induction is about change in the magnetic situation at a loop. But “magnetic situation” is not a precise phrase. To write any quantitative law, we need a single number that captures how much magnetic field is threading through a given surface. That number is the magnetic flux.

The Physical Idea

Imagine holding a wire loop in a stream of water. The amount of water flowing through the loop per second depends on three things: how fast the water moves, how large the loop is, and how the loop is oriented relative to the flow. A loop held face-on to the current intercepts the maximum flow. The same loop tilted almost edge-on to the current intercepts very little, even in the same stream. Tilt it to exactly edge-on and the net flow through it is zero.

Magnetic flux is the direct analogue for the magnetic field. It measures how much of the magnetic field passes through a chosen surface.

The Area Vector

To make this quantitative, we assign the surface a vector character. For a flat surface of area \(A\), we define the area vector \(\vec{A}\) as a vector whose magnitude is \(A\) and whose direction is perpendicular to the surface (along the surface's normal).

For a closed surface there is a natural choice: the outward normal. For an open surface such as a wire loop, there are two possible normals (one on each side). The choice of which normal to call positive is a convention; what matters is that the same choice is used consistently throughout a calculation. For a flat coil, the right-hand rule provides the standard convention: curl the fingers of the right hand in the direction of the chosen positive current flow, and the thumb points along the positive normal.

Core intuition: Area Vector and Orientation

The area vector \(\vec{A}\) encodes both the size of the surface and its orientation in space. Two loops of the same physical area have the same \(|\vec{A}|\), but if they are tilted differently in a magnetic field, they present different amounts of area to the field — and therefore intercept different amounts of flux. Flux is fundamentally about perpendicular throughness: only the component of \(\vec{B}\) that pierces the surface (i.e. the component parallel to \(\hat{n}\)) contributes. The component of \(\vec{B}\) that lies in the plane of the surface glides along it and contributes nothing.

Definition of Magnetic Flux

For a uniform magnetic field \(\vec{B}\) over a flat surface with area vector \(\vec{A}\), the magnetic flux \(\Phi_B\) through that surface is defined as

\[ \Phi_B \;=\; \vec{B}\cdot\vec{A} \;=\; BA\cos\theta, \]

where \(\theta\) is the angle between \(\vec{B}\) and the area vector \(\hat{n}\) (the surface normal).

The SI unit of magnetic flux is the weber (Wb): \[ 1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2. \]

Understanding the Angle \(\theta\)

The angle \(\theta\) in equation is between \(\vec{B}\) and the normal to the surface — not between \(\vec{B}\) and the surface itself. This distinction is frequently confused in examinations.

\(\theta = 0\): field is perpendicular to the surface (parallel to the normal). Maximum flux: \(\Phi_B = BA\).
\(\theta = 90°\): field lies in the plane of the surface (perpendicular to the normal). Zero flux: \(\Phi_B = 0\).
\(\theta = 180°\): field is antiparallel to the normal. Flux is \(-BA\) (negative sign indicates field threads from the “back” side of the chosen normal).

Misconception alert: “Field Lines Near the Loop Are Flux”

“More field lines passing near the loop means more flux through it.” Flux measures how many field lines pass through the surface, not how many pass near it. A very strong field directed parallel to the plane of the loop produces zero flux through it, no matter how intense the field is. Conversely, even a weak field directed exactly perpendicular to the loop produces a nonzero flux. A related confusion: students sometimes treat magnetic flux as literally “counting field lines” in a quantitative sense. Field lines are a visual aid; flux is the precise quantity. \(\Phi_B = BA\cos\theta\) is exact; counting field-line sketches is not.

Non-Uniform Fields and Curved Surfaces

When the field is not uniform, or the surface is not flat, the flux is found by dividing the surface into infinitesimal area elements \(d\vec{A}\) and integrating: \[ \Phi_B = \iint_S \vec{B}\cdot d\vec{A}. \] For the problems in this chapter — and for almost all JEE and NEET situations — the field is uniform and the surface is a flat loop, so equation applies directly.

✓ Checkpoint · Magnetic Flux

A circular loop of radius \(r = 0.15\,\text{m}\) is placed perpendicular to a uniform magnetic field \(B = 0.80\,\text{T}\) (i.e., the field is parallel to the normal to the loop). Calculate the magnetic flux through the loop.
\(\Phi_B = B\pi r^2 = 0.80\times\pi\times(0.15)^2 \approx 0.057\,\text{Wb}\).
The same loop is now rotated so that its plane is parallel to the field (the field lies in the plane of the loop). What is the flux now?
Zero. The field is perpendicular to the area vector (\(\theta=90°\)), so \(\Phi_B = BA\cos90°=0\).
A square loop of side \(a\) is placed in a uniform field \(B\). The angle between the field and the plane of the loop is \(\alpha\) (note: angle with the plane, not the normal). Write an expression for the flux.
The angle between \(\vec{B}\) and the normal is \(90°-\alpha\), so \(\Phi_B = Ba^2\cos(90°-\alpha) = Ba^2\sin\alpha\). Be careful: the given angle is with the plane; the flux formula uses the angle with the normal.
Two identical loops are placed in the same uniform magnetic field. Loop A has its normal parallel to \(\vec{B}\). Loop B has its normal at \(45°\) to \(\vec{B}\). What is the ratio \(\Phi_A : \Phi_B\)?
\(\Phi_A : \Phi_B = BA\cos0°\,:\,BA\cos45° = 1 : \tfrac{1}{\sqrt{2}} = \sqrt{2}:1\).

Faraday's Law of Electromagnetic Induction

◈ Faraday & Lenz bench

live bench

Drag the magnet through the coil (or auto-sweep) — live Φ and ε = −dΦ/dt strip-charts, the galvanometer needle kicks, and the induced-current arrows flip sign with the Lenz verdict.

◈ loading the interactive bench…

↗ open in the Simulations page

We now have the right vocabulary. Section established that induction depends on change. Section gave us the precise quantity — magnetic flux \(\Phi_B\) — that captures what is changing. The question that remains is quantitative: how exactly does the induced emf depend on the change in flux?

From Experiment to Law

Faraday's experiments revealed a clean pattern. When a loop is placed in a changing magnetic environment:

The induced emf is larger when the flux changes faster.
The induced emf is larger when there are more turns of wire (each turn adds its contribution).
The direction of the induced emf reverses when the flux is decreasing rather than increasing.

The first two observations point immediately to a law involving the rate of change of flux, scaled by the number of turns. The third observation is the seed of Lenz's law, which will be developed in the next section.

Quantitatively, Faraday's law states:

\[ \mathcal{E} = -N\,\frac{d\Phi_B}{dt}, \]

where \(\mathcal{E}\) is the induced emf, \(N\) is the number of turns in the coil, and \(d\Phi_B/dt\) is the rate of change of the magnetic flux through one turn of the coil.

The negative sign is the mathematical encoding of Lenz's law — it will be treated carefully in Section . For computing magnitudes:

\[ |\mathcal{E}| = N\left|\frac{d\Phi_B}{dt}\right|. \]

Birth of the formula: Why the Rate of Change?

Consider a loop in which the flux is changing at a rate \(d\Phi_B/dt\). Over a small time interval \(dt\), the flux changes by \(d\Phi_B\). This change in flux through the loop is what drives the induced emf. Experimentally (and this is a postulate, not derivable from earlier mechanics), the induced emf is directly proportional to \(d\Phi_B/dt\). For a coil of \(N\) turns, each turn contributes an emf of \(-d\Phi_B/dt\) (assuming the same flux passes through each turn — a good approximation for a tightly wound coil). The turns are connected in series, so their emfs add: \[ \mathcal{E} = -N\,\frac{d\Phi_B}{dt}. \] The quantity \(N\Phi_B\) is called the flux linkage \(\lambda\), and Faraday's law can be written compactly as \(\mathcal{E} = -d\lambda/dt\).

What “Rate of Change of Flux” Means Physically

The power of equation is that \(d\Phi_B/dt\) can be nonzero for entirely different physical reasons, and the law applies to all of them:

Changing \(B\), fixed loop. \(\Phi_B = BA\cos\theta\). If \(B\) changes with time while \(A\) and \(\theta\) remain fixed: \(d\Phi_B/dt = A\cos\theta\;(dB/dt)\).
Changing area, fixed \(B\). If the loop is deformable and its area changes — for instance, one side of a rectangular loop slides along rails in a field — then \(d\Phi_B/dt = B\cos\theta\;(dA/dt)\).
Changing angle, fixed \(B\) and fixed area. If the loop rotates in a steady field, \[ \frac{d\Phi_B}{dt} = -BA\sin\theta\;\frac{d\theta}{dt}. \] This is the operating principle of every AC generator.
Relative motion. A loop moving into or out of a non-uniform field region experiences a changing flux even if neither \(B\) nor the loop's area is altered in isolation.

Faraday's law handles all four cases with a single equation. This is one of the great unifying achievements of classical electromagnetism.

Core intuition: Resolution of Perplexing Question 4

Here is the unification the chapter opened with. A power-station generator (changing angle), a train's eddy-current brake (relative motion of conductor and field), and a phone charger's transformer (changing current in a coupled coil) look like utterly different machines. Yet each is nothing but a way of making \(d\Phi_B/dt \neq 0\), and every one obeys the single equation \(\mathcal{E} = -N\,d\Phi_B/dt\). The device changes; the law does not.

Misconception alert: “Induction Requires a Moving Magnet”

“Faraday's law applies only when a magnet is moving near a coil.” This is a serious over-restriction. Students who have only seen the “moving magnet” demonstration sometimes believe that physical motion of a source is necessary for induction. Faraday's law depends on \(d\Phi_B/dt\), not on the mechanism that produces the change. A fixed coil in a time-varying field (produced, for instance, by a nearby electromagnet with changing current) experiences exactly the same induction. A rotating coil in a steady field experiences induction. A loop whose shape is changing in a steady field experiences induction. The only requirement is that the magnetic flux through the loop is changing.

A Note on the Negative Sign and What Comes Next

Equation carries a negative sign. This sign is not a mathematical accident. It encodes a fundamental physical constraint: the induced emf acts in a direction that opposes the change in flux that caused it. This is Lenz's law, and it deserves its own careful treatment.

The distinction between computing the magnitude of the induced emf (Section , and all examples above) and determining its direction (Lenz's law, Section ) is one that examination problems deliberately probe. For now, use the magnitude form and reserve the sign for the next section.

✓ Checkpoint · Faraday's Law of Electromagnetic Induction

The magnetic flux through a single-turn coil changes from \(0.12\,\text{Wb}\) to \(0.04\,\text{Wb}\) in \(0.080\,\text{s}\). Find the magnitude of the induced emf.
\(|\mathcal{E}| = |{\Delta\Phi_B}/{\Delta t}| = {(0.12-0.04)}/{0.080} = 1.0\,\text{V}\).
A 500-turn coil has a cross-sectional area of \(4.0\times10^{-3}\,\text{m}^2\). The magnetic field through it (perpendicular to the coil) decreases at a uniform rate of \(0.20\,\text{T\,s}^{-1}\). Find the induced emf.
\(|d\Phi_B/dt| = A\,|dB/dt| = 4.0\times10^{-3}\times0.20 = 8.0\times10^{-4}\,\text{Wb\,s}^{-1}\). \(|\mathcal{E}| = N\,|d\Phi_B/dt| = 500\times8.0\times10^{-4} = 0.40\,\text{V}\).
A conducting loop of area \(A\) is placed in a field \(B(t) = B_0\sin(\omega t)\). The loop's normal is parallel to the field. Derive an expression for the induced emf as a function of time. At what instants is the emf maximum? At what instants is it zero?
\(\Phi_B = B_0 A\sin(\omega t)\); \(\mathcal{E} = -A\,d(B_0\sin\omega t)/dt = -AB_0\omega\cos(\omega t)\). Magnitude maximum when \(|\cos(\omega t)|=1\), i.e. \(\omega t = 0,\,\pi,\,2\pi,\ldots\) (when the field is zero but changing fastest). Emf is zero when \(\cos(\omega t)=0\), i.e. when the field is at its maximum or minimum (changing most slowly).
A rectangular loop of width \(w\) and length \(\ell\) has one side sliding at speed \(v\) in a uniform field \(B\) (perpendicular to the loop). Express the induced emf in terms of \(B\), \(w\), and \(v\) only. Why does \(\ell\) not appear in the answer?
\(|\mathcal{E}| = Bwv\). The rate of area change is \(dA/dt = wv\) (only the width \(w\) and speed \(v\) of the sliding side matter); the total length \(\ell\) of the loop determines the current value of the area but does not affect its rate of change.

Solved examples

Five fully-worked problems from this chapter, free — solution and answer shown in full. The complete set of worked examples is in the full book.

Example 1 · Qualitative Analysis: Magnet–Coil Interaction

A bar magnet is held with its north pole facing a flat circular coil. The magnet is then pushed slowly toward the coil. Describe: (a) whether a current is induced in the coil, (b) what happens if the magnet is brought to rest close to — but not inside — the coil, and (c) what changes if the magnet is subsequently pulled away rapidly.

Solution. (a) Magnet approaching. As the north pole moves toward the coil, the magnetic field threading through the coil increases. The magnetic situation is changing, so an induced emf appears and drives an induced current through the coil. The direction of this current (determined by Lenz's law) is such as to oppose the increasing field. (b) Magnet held stationary. Once the magnet stops, the field through the coil is no longer changing. The induced emf drops to zero immediately, and the induced current ceases. The field is still present and still threads the coil, but a static field produces no induction. (c) Magnet pulled away rapidly. The field through the coil now decreases, and faster than before because the motion is rapid. An induced current reappears, this time in the direction opposite to case (a) because the field is decreasing rather than increasing. The faster the withdrawal, the larger the induced emf — the key quantity is the rate of change, not merely whether a change occurs. Check. The three parts are internally consistent: current flows only while the flux is changing, and it reverses sign when the change reverses (approach vs.\ withdrawal). Part (b) — the stationary magnet giving exactly zero — is the limiting case that anchors the whole picture: no rate of change, no emf.

Example 2 · Flux Through a Tilted Rectangular Loop

A rectangular loop of dimensions \(0.20\,\text{m}\times 0.30\,\text{m}\) is placed in a uniform magnetic field of magnitude \(0.50\,\text{T}\). The normal to the loop makes an angle of \(60°\) with the field direction. Find the magnetic flux through the loop.

Solution. The area of the loop is \(A = 0.20 \times 0.30 = 0.060\,\text{m}^2\), and the angle between \(\vec{B}\) and the area vector is \(\theta = 60°\). Therefore, \[ \Phi_B = BA\cos\theta = 0.50 \times 0.060 \times \cos 60° = 0.030 \times 0.5 = 0.015\,\text{Wb}. \] Physical interpretation. Only the component of \(\vec{B}\) perpendicular to the loop — \(B\cos 60° = 0.25\,\text{T}\) — contributes to the flux. The remaining component (\(B\sin 60°\)) lies in the plane of the loop and passes along the surface rather than through it. Check. Limiting cases bracket the answer: \(\theta = 0\) would give the maximum \(BA = 0.030\,\text{Wb}\) and \(\theta = 90°\) would give \(0\). Our \(60°\) result sits between them at exactly half the maximum, as it must since \(\cos 60° = \tfrac12\). Units: \(\text{T}\cdot\text{m}^2 = \text{Wb}\). ✓

Example 3 · Zero Flux from a Strong Field

A square loop of side \(0.10\,\text{m}\) lies flat on a horizontal table. A horizontal magnetic field of magnitude \(2.0\,\text{T}\) is directed parallel to the table surface. What is the magnetic flux through the loop?

Solution. The loop lies in the horizontal plane, so its normal points vertically. The field is horizontal — it makes a \(90°\) angle with the normal. Therefore \(\Phi_B = BA\cos 90° = 0\). Despite the field being strong (\(2.0\,\text{T}\)) and present at the loop's location, no flux passes through the loop: the field glides along the surface without piercing it. This is also why this configuration produces no induction, no matter how the field magnitude changes. Check. Contrast with the opposite extreme: had the loop been turned so its normal aligned with \(\vec{B}\), the flux would be the maximum \(BA = 2.0 \times 0.010 = 0.020\,\text{Wb}\). The stated edge-on orientation is the other end of the range, \(\Phi_B = 0\) — and the field strength never enters a \(\theta = 90°\) result. ✓

Example 4 · Induced EMF from a Changing Field

A circular coil of 200 turns and radius \(0.10\,\text{m}\) is placed in a uniform magnetic field perpendicular to the plane of the coil. The field increases uniformly from \(0.20\,\text{T}\) to \(0.80\,\text{T}\) in \(0.50\,\text{s}\). Find the magnitude of the induced emf.

Solution. The area of one turn is \(A = \pi r^2 = \pi \times 10^{-2}\,\text{m}^2\). Since the field is perpendicular to the coil, \(\theta = 0\) and \(\Phi_B = BA\) per turn. The rate of change of flux is \[ \frac{d\Phi_B}{dt} = A\,\frac{dB}{dt} = \pi \times 10^{-2} \times \frac{0.80 - 0.20}{0.50} = 1.2\pi \times 10^{-2}\,\text{Wb\,s}^{-1}. \] By Faraday's law (magnitude), \[ |\mathcal{E}| = N\left|\frac{d\Phi_B}{dt}\right| = 200 \times 1.2\pi \times 10^{-2} = 2.4\pi \approx 7.54\,\text{V}. \] Physical interpretation. A single turn would give only about \(0.038\,\text{V}\); the factor \(N = 200\) multiplies it to \(7.5\,\text{V}\). This multiplicative role of \(N\) is why real transformers and generators use multi-turn coils. Check. Independent route through the single-turn emf: \(A|dB/dt| = \pi\times10^{-2}\times1.2 = 0.0377\,\text{V}\), and \(200 \times 0.0377 = 7.54\,\text{V}\) — agreeing. Units: \(\text{m}^2\cdot\text{T\,s}^{-1} = \text{Wb\,s}^{-1} = \text{V}\). ✓

Example 5 · Direction of Induced Current: Approaching Magnet

A bar magnet is held with its north pole pointing downward, directly above a horizontal circular loop lying flat on a table. The magnet is moved downward toward the loop. Determine the direction of the induced current in the loop as viewed from above.

Solution. Step 1 — Flux direction. The north pole points downward, so field lines pass downward through the loop; the external flux points down (into the table). Step 2 — Flux change. As the magnet approaches, the field strengthens: the downward flux is increasing. Step 3 — Required induced field. By Lenz's law, the induced current must create a field through the loop that opposes the increase — pointing upward (out of the table). Step 4 — Current direction. By the right-hand rule, a field pointing up through the loop requires current flowing counterclockwise as viewed from above. Physical interpretation. The counterclockwise current makes the top face of the loop a north pole, which repels the approaching north pole of the magnet — opposing the motion, consistent with Lenz's law and energy conservation. Check. Energy test: because the induced north pole repels the approaching magnet, the agent must do positive work to keep pushing — that work is the source of the \(I^2R\) heat in the loop. The opposite (clockwise) answer would attract the magnet and deliver energy for free, violating conservation. ✓

Problem bank

Five questions from this chapter’s 50-question bank, free — attempt each one before you reveal the answer. The rest of the bank, and the timed test that draws on all of it, are in the full book.

✎ Free practice questions

When Does Induction Occur?
State whether an emf is induced and give a one-sentence reason. (a) A rectangular loop at rest in a uniform static field. (b) The same loop translated (no rotation) in a uniform field. (c) The loop moved from a strong-field region to a weak-field region, field always perpendicular to the loop. (d) The loop stationary but the field oscillating sinusoidally.
(a) No: flux is constant. (b) No: translating in a uniform field does not change flux. (c) Yes: flux decreases as the loop enters the weaker region. (d) Yes: \(d\Phi_B/dt\ne0\) at all times except the flux extrema.
Flux and Angle
A square loop of side \(a = 0.20\,\text{m}\) is placed in uniform field \(B = 0.60\,\text{T}\). (a) Flux when field is perpendicular to the loop plane. (b) Flux when field makes \(30°\) with the loop plane (\(60°\) with normal). (c) What orientation gives zero flux?
(a) \(\Phi = BA = 0.024\,\text{Wb}\). (b) \(\Phi = BA\cos60° = 0.012\,\text{Wb}\). (c) Field lying in the plane of the loop (\(\theta = 90°\) with normal).
Faraday Magnitude
A 50-turn coil of area \(6.0\times10^{-3}\,\text{m}^2\) is in a field perpendicular to the coil, decreasing uniformly from \(0.80\,\text{T}\) to \(0.20\,\text{T}\) in \(0.15\,\text{s}\). Find the induced emf.
\(|d\Phi_B/dt| = 6\times10^{-3}\times(0.60/0.15) = 0.024\,\text{Wb\,s}^{-1}\); \(|\mathcal{E}| = 50\times0.024 = 1.2\,\text{V}\).
Sliding Rod with Resistance
Rails \(\ell = 0.60\,\text{m}\) apart; external \(R = 3.0\,\Omega\); rod of negligible resistance slides at \(v = 5.0\,\text{m\,s}^{-1}\); \(B = 0.25\,\text{T}\) perpendicular to the rail plane. Find: (a) emf, (b) current, (c) power in \(R\), (d) force to maintain speed.
(a) \(0.75\,\text{V}\). (b) \(0.25\,\text{A}\). (c) \(0.1875\,\text{W}\). (d) \(F = BI\ell = 0.0375\,\text{N}\); check: \(Fv = 0.1875\,\text{W}\). ✓
Mutual vs. Self: Comparing Induced EMFs
\(M = 0.050\,\text{H}\); \(L_1 = 0.30\,\text{H}\); \(L_2 = 0.10\,\text{H}\); \(dI_1/dt = 8.0\,\text{A\,s}^{-1}\). (a) Emf in coil 2 due to mutual induction. (b) Back emf in coil 1 due to self-induction. (c) Which is larger and by what factor? (d) Coupling coefficient \(k = M/\sqrt{L_1 L_2}\).
(a) \(0.40\,\text{V}\). (b) \(2.4\,\text{V}\). (c) Self-induction emf larger by factor 6. (d) \(k = 0.050/\sqrt{0.030} \approx 0.289\).