Faraday's Laws of Electromagnetic Induction
Magnetic Flux — A Quick Recap
Before discussing induction, recall that magnetic flux through a surface is $\Phi_B = BA\cos\theta$, where $\theta$ is the angle between the field and the surface's normal — flux can change due to a changing field strength, a changing area, or a changing orientation angle, and electromagnetic induction can arise from a change in any one of these three factors.
First Law
Whenever the magnetic flux linked with a circuit changes, an EMF is induced in it — this induced EMF lasts only as long as the flux is actually changing.
Second Law
The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux:
$$\varepsilon = -\dfrac{d\Phi_B}{dt}$$
For a coil of $N$ turns: $\varepsilon = -N\dfrac{d\Phi_B}{dt}$.
Solved Example
A coil of 100 turns experiences a change in magnetic flux from 0.01 Wb to 0.05 Wb over 0.2 s. Calculate the induced EMF.
$\varepsilon = N\dfrac{\Delta\Phi}{\Delta t} = (100)\dfrac{0.05-0.01}{0.2} = (100)\dfrac{0.04}{0.2} = (100)(0.2) = 20\ \text{V}$
Q. Why is no EMF induced in a coil sitting in a magnetic field that is strong but remains perfectly constant over time?
Ans: Faraday's law states that EMF is induced only when magnetic flux changes over time — a constant field, even a very strong one, produces a constant (unchanging) flux through the coil, meaning dΦ/dt = 0. Since the induced EMF depends specifically on this rate of change, not on the absolute strength of the flux itself, no EMF is induced regardless of how strong the constant field is.
Q. A single-turn coil experiences a flux change of 0.02 Wb in 0.1 s. If the coil instead had 50 turns (same flux change per turn), how would the induced EMF compare?
Ans: EMF for 1 turn = 0.02/0.1 = 0.2 V. EMF for 50 turns = 50 × 0.2 = 10 V — the induced EMF is 50 times larger, directly proportional to the number of turns.
Solved Example
Give three distinct physical ways in which the magnetic flux through a coil can be changed to induce an EMF, based on the flux formula Φ = BAcosθ.
(1) Changing the magnetic field strength B — e.g., moving a magnet closer to or farther from the coil, or varying the current in a nearby electromagnet.
(2) Changing the area A of the coil exposed to the field — e.g., expanding or shrinking a flexible loop, or a sliding rod changing the enclosed circuit area.
(3) Changing the angle θ between the coil's plane and the field — e.g., rotating the coil relative to a fixed magnetic field, as in a generator.
Q. A coil is rotated within a constant magnetic field, with its area and the field strength both unchanged. Explain how an EMF can still be induced.
Ans: Even though B and A remain constant, rotating the coil changes the angle θ between the coil's plane (or its normal) and the magnetic field — since flux Φ = BAcosθ depends on this angle, a changing θ produces a changing flux over time, which by Faraday's law induces an EMF, exactly as occurs in the rotating coil of an AC generator.
Lenz's Law
Statement
The direction of an induced current is always such that it opposes the very change in magnetic flux that produced it — this is the physical origin of the negative sign in Faraday's law, and is itself a direct consequence of the law of conservation of energy.
Solved Example
Explain why the induced current, when a bar magnet's north pole is pushed towards a coil, flows in a direction that creates a north pole facing the approaching magnet.
By Lenz's law, the induced current must oppose the increasing flux caused by the approaching magnet. Creating a north pole on the coil's near face means the coil's induced magnetic field pushes back against (repels) the approaching magnet's north pole — this opposition is precisely what resists the change in flux (the magnet's approach), consistent with Lenz's law, and also explains why work must be done (against this repulsion) to push the magnet closer, which is where the electrical energy delivered to the circuit actually comes from.
Q. Why is Lenz's law considered a direct consequence of the law of conservation of energy?
Ans: If the induced current instead flowed in a direction that assisted (rather than opposed) the change in flux, it would create a self-reinforcing effect — the induced current would strengthen the changing flux, which would induce even more current, indefinitely amplifying itself with no external energy input, creating energy from nothing. Since this violates energy conservation, the induced current must instead oppose the change, requiring external work (energy) to be done to maintain or continue that change, which is exactly what Lenz's law states.
Q. A coil experiences a decreasing magnetic flux (field weakening) through it. In what direction will the induced current flow, according to Lenz's law?
Ans: The induced current will flow in a direction that tries to maintain (oppose the decrease of) the original flux — meaning it will flow in the same direction as would create a magnetic field reinforcing the original, weakening field, effectively trying to "resist" the flux's decrease.
Motional EMF
Definition
A motional EMF is induced when a conductor moves through a magnetic field, causing free charges within it to experience a magnetic force, redistributing until an equilibrium (induced EMF) is established.
Solved Example
A metal rod of length 0.5 m moves with a velocity of 4 m/s perpendicular to a magnetic field of 0.2 T. Calculate the motional EMF induced.
$\varepsilon = Bvl = (0.2)(4)(0.5) = 0.4\ \text{V}$
Q. Why is a motional EMF considered a special case of Faraday's law, rather than an entirely separate phenomenon?
Ans: When a rod moves through a magnetic field, it sweeps out an increasing area, and this changing area (within the field) causes a continuously changing magnetic flux through any circuit the rod is part of — this is exactly the situation Faraday's law describes. The formula ε = Bvl can, in fact, be derived directly from ε = −dΦ/dt by considering the rate of change of the swept area, showing that motional EMF is simply Faraday's law applied to this specific, physically intuitive scenario of a moving conductor.
Q. If the velocity of a rod moving through a magnetic field is doubled, how does the motional EMF change?
Ans: Since ε ∝ v, doubling the velocity doubles the motional EMF.
Eddy Currents
Definition
Eddy currents are circulating (loop-like) induced currents that arise within the bulk of a conducting material whenever it experiences a changing magnetic flux — unlike current confined to a wire, these currents can flow in any direction within the material's volume, forming closed loops that oppose the flux change (per Lenz's law).
| Application | How Eddy Currents Are Used |
|---|
| Electromagnetic braking | Eddy currents induced in a moving metal part oppose its motion, providing smooth, contact-free braking (used in some trains and roller coasters) |
| Induction furnaces | Eddy currents generate intense heat directly within a metal, used for melting metals without any direct contact |
| Energy loss in transformers | Unwanted eddy currents in a transformer's iron core cause energy loss as heat, minimised by using laminated (thin, insulated) sheets instead of a solid core |
Solved Example
Explain why transformer cores are made of thin, laminated sheets of iron (insulated from each other) rather than a single solid block of iron.
A solid iron core, being a good conductor, would allow large eddy currents to circulate freely within its bulk in response to the constantly changing magnetic flux (due to the AC current), generating significant unwanted heat and wasting energy. Laminating the core into thin sheets, each electrically insulated from its neighbours, breaks up these large current loops into many much smaller ones confined within each thin sheet — since eddy current magnitude depends on the area available for current to circulate, this significantly reduces the eddy currents (and hence energy loss) compared to a single solid block.
Q. Why do eddy currents provide a smooth, gradual braking effect (rather than an abrupt stop) when used in electromagnetic braking systems?
Ans: The magnitude of induced eddy currents (and hence the opposing braking force) depends directly on the rate of change of flux, which in turn depends on the speed of the moving conductor — at higher speeds, stronger eddy currents (and stronger braking force) are induced, while as the object slows down, the induced currents (and braking force) naturally weaken as well. This speed-dependent, self-adjusting braking force provides a smooth, gradually diminishing deceleration rather than an abrupt, constant-force stop.
Q. Name one useful application and one undesirable consequence of eddy currents.
Ans: Useful application: induction furnaces (or electromagnetic braking). Undesirable consequence: energy loss (heating) in transformer cores, which is why laminated cores are used to minimise this effect.
Q. Why do eddy current losses in a transformer core increase significantly if the core is made of a single thick block rather than thin laminated sheets, even though the total volume of iron is the same?
Ans: Eddy currents circulate within continuous conducting loops of material, and the induced EMF driving these currents depends on the area enclosed by each loop — a single thick block allows large, continuous current loops to form throughout its bulk, and larger enclosed loop areas support proportionally larger induced eddy currents (and hence greater I²R heating losses). Thin, insulated laminations restrict each loop to a much smaller cross-sectional area within an individual thin sheet, since the insulation between sheets prevents current from flowing between them, substantially reducing the eddy current magnitude and associated energy loss for the same total iron volume.
Self-Inductance
Definition
Self-inductance (L) is a coil's property of opposing any change in the current flowing through itself, via the EMF it induces in response to this self-generated changing flux: $\Phi_B = LI$, and $\varepsilon = -L\dfrac{dI}{dt}$.
Solved Example
A coil has a self-inductance of 0.5 H. If the current through it changes at a rate of 4 A/s, calculate the induced EMF.
$\varepsilon = L\dfrac{dI}{dt} = (0.5)(4) = 2\ \text{V}$
Q. Why is self-inductance often described as the electrical analogue of mechanical inertia?
Ans: Just as mechanical inertia resists any change in an object's velocity (requiring a force to speed up or slow down), self-inductance resists any change in the current flowing through a coil — inducing an opposing EMF whenever current tries to increase or decrease, effectively "fighting" against any change in the electrical state of the circuit, exactly analogous to how inertia resists changes in mechanical motion.
Q. If the number of turns in a solenoid is doubled (all other dimensions unchanged), how does its self-inductance change?
Ans: Since L ∝ N², doubling the number of turns increases self-inductance by a factor of 2² = 4.
Energy Stored in an Inductor
Just as a capacitor stores energy in its electric field, an inductor stores energy in its magnetic field when carrying a current:
$$U = \tfrac{1}{2}LI^2$$
Solved Example
Calculate the energy stored in a 0.4 H inductor carrying a current of 3 A.
$U = \tfrac{1}{2}LI^2 = \tfrac{1}{2}(0.4)(3)^2 = \tfrac{1}{2}(0.4)(9) = 1.8\ \text{J}$
Q. Why is a sudden interruption of current in a highly inductive circuit (like an electromagnet) potentially dangerous, in terms of the energy stored in the inductor?
Ans: The inductor's stored magnetic energy (½LI²) must go somewhere when the circuit is suddenly broken — since the inductor strongly opposes any sudden change in current, it induces a very large back-EMF in an attempt to maintain the current flow, and this can produce a dangerously high voltage spike (potentially causing sparking or arcing across the switch) as the stored energy is rapidly released, which is why circuits with large inductors often include protective components to safely dissipate this energy.
Mutual Inductance
Definition
Mutual inductance (M) describes how a changing current in one coil induces an EMF in a nearby, separate (but magnetically coupled) coil: $\varepsilon_2 = -M\dfrac{dI_1}{dt}$.
Solved Example
Two coils have a mutual inductance of 0.2 H. If the current in the primary coil changes at a rate of 5 A/s, calculate the EMF induced in the secondary coil.
$\varepsilon_2 = M\dfrac{dI_1}{dt} = (0.2)(5) = 1\ \text{V}$
Q. Why does bringing two coils closer together generally increase their mutual inductance?
Ans: Mutual inductance depends on how much of the magnetic flux produced by one coil actually passes through (links with) the other coil. Bringing the coils closer together increases this flux linkage — more of the field lines from the first coil pass through the area of the second coil — resulting in a stronger magnetic coupling between them and hence a larger mutual inductance.
Q. What is the fundamental difference between self-inductance and mutual inductance?
Ans: Self-inductance describes how a coil's own changing current induces an EMF within itself (due to its own changing flux). Mutual inductance describes how a changing current in one coil induces an EMF in a separate, different coil, due to the magnetic flux linkage between the two distinct circuits.
AC Generator
Principle
An AC generator converts mechanical energy into electrical energy, based on electromagnetic induction — a coil is mechanically rotated within a magnetic field (or a magnetic field is rotated relative to a stationary coil), continuously changing the flux through the coil and generating an alternating (sinusoidally varying) EMF.
Solved Example
A coil of 200 turns, area 0.05 m², rotates at an angular frequency of 100 rad/s in a magnetic field of 0.3 T. Calculate the peak EMF generated.
$\varepsilon_0 = NBA\omega = (200)(0.3)(0.05)(100) = 300\ \text{V}$
Q. Why does an AC generator produce a sinusoidally varying EMF, rather than a constant one, as the coil rotates at constant angular speed?
Ans: As the coil rotates at constant angular velocity, the rate of change of magnetic flux through it (which determines the instantaneous induced EMF) varies sinusoidally — the flux itself is maximum when the coil is perpendicular to the field (rate of change momentarily zero there) and zero when the coil is parallel to the field (rate of change maximum there). This continuously varying rate of flux change, following a sine/cosine pattern as the coil rotates uniformly, is exactly why the generated EMF itself is sinusoidal.
Q. At what orientation of the coil (relative to the magnetic field) is the instantaneous EMF generated by an AC generator maximum?
Ans: When the plane of the coil is parallel to the magnetic field (i.e., the coil's normal is perpendicular to B) — at this orientation, the flux through the coil is momentarily zero, but the rate of change of flux (and hence induced EMF) is at its maximum, since the coil is "cutting through" field lines most rapidly at this point in its rotation.
AC Circuits — Resistor, Inductor & Capacitor
| Element | Reactance | Phase Relationship (Voltage vs Current) |
|---|
| Resistor | R (unchanged with frequency) | In phase (0° difference) |
| Inductor | $X_L = \omega L$ | Voltage leads current by 90° |
| Capacitor | $X_C = \dfrac{1}{\omega C}$ | Voltage lags current by 90° |
Memory Aid — "ELI the ICE man"
In an inductor (L), EMF (E) leads current (I) — "ELI." In a capacitor (C), current (I) leads EMF (E) — "ICE." This simple mnemonic helps remember the phase relationships for inductive and capacitive AC circuits.
Solved Example
Calculate the inductive reactance of a 0.2 H inductor in an AC circuit with angular frequency 100 rad/s.
$X_L = \omega L = (100)(0.2) = 20\ \Omega$
Q. Why does the reactance of an inductor increase with frequency, while the reactance of a capacitor decreases with frequency?
Ans: An inductor opposes any change in current (via self-induced EMF), and at higher frequencies, current changes more rapidly — so the inductor's opposition (reactance) increases with frequency (X_L = ωL). A capacitor's opposition arises from how much it can charge/discharge within each cycle; at higher frequencies, there's less time per cycle for charge to build up on its plates, so less opposition (voltage buildup) is needed, meaning capacitive reactance decreases with increasing frequency (X_C = 1/ωC).
Q. Calculate the capacitive reactance of a 50 μF capacitor at an angular frequency of 200 rad/s.
Ans: X_C = 1/(ωC) = 1/((200)(50×10⁻⁶)) = 1/0.01 = 100 Ω.
RMS Values of AC
Since alternating current continuously varies (and averages to zero over a full cycle), a useful equivalent measure is the root mean square (RMS) value — the value of an equivalent steady DC that would produce the same average heating (power dissipation) effect: $I_{rms} = \dfrac{I_0}{\sqrt{2}}$, $V_{rms} = \dfrac{V_0}{\sqrt{2}}$, where $I_0, V_0$ are peak values. Household AC voltage ratings (like 220 V or 110 V) refer to RMS values, not peak values.
Solved Example
An AC supply has a peak voltage of 311 V. Calculate its RMS voltage, and explain why this closely matches typical household voltage ratings.
$V_{rms} = \dfrac{V_0}{\sqrt{2}} = \dfrac{311}{1.414} \approx 220\ \text{V}$
This matches the standard 220 V household AC rating, confirming that quoted household voltages are RMS values, not the actual (higher) peak voltage the supply reaches during each cycle.
Q. Why is the RMS value, rather than the peak value or simple average value, used to specify practical AC voltage and current ratings?
Ans: The simple average of a sinusoidal AC quantity over a full cycle is exactly zero (since it's equally positive and negative), making it useless for describing the practical, usable strength of the supply. The RMS value, however, is specifically defined so that it produces the same average heating/power effect as an equivalent steady DC value would — this makes RMS values directly meaningful and comparable for practical purposes like power calculations and appliance ratings, which is why they are the standard convention for quoting AC voltage and current.
LCR Series Circuit & Resonance
Impedance
For a series LCR circuit, the combined opposition to current (impedance, $Z$) accounts for the phase differences between R, L, and C: $Z = \sqrt{R^2+(X_L-X_C)^2}$.
Resonance
Resonance occurs when $X_L = X_C$, at which point impedance is at its minimum (equal to just $R$), and current through the circuit is at its maximum.
Resonant Angular Frequency
$$\omega_0 = \dfrac{1}{\sqrt{LC}}$$
Solved Example
Calculate the resonant frequency of an LCR circuit with L = 0.5 H and C = 20 μF.
$\omega_0 = \dfrac{1}{\sqrt{LC}} = \dfrac{1}{\sqrt{(0.5)(20\times10^{-6})}} = \dfrac{1}{\sqrt{1\times10^{-5}}} = \dfrac{1}{0.00316} \approx 316.2\ \text{rad/s}$
Q. Why is the current maximum at resonance in a series LCR circuit?
Ans: At resonance, X_L exactly equals X_C, so the term (X_L−X_C) in the impedance formula becomes zero, leaving impedance Z equal to just R (its minimum possible value, since resistance alone is the smallest Z can be). Since current I = V/Z, this minimum impedance corresponds to the maximum possible current for a given applied voltage, which is exactly the resonance condition.
Q. If the inductance in an LCR circuit is doubled while capacitance is halved, how does the resonant frequency change?
Ans: Since ω₀ = 1/√(LC), and the product LC remains unchanged (doubling L while halving C keeps LC the same), the resonant frequency remains exactly the same as before.
Power in an AC Circuit — Power Factor
Unlike a DC circuit, average power in an AC circuit depends on the phase difference ($\phi$) between voltage and current:
$$P_{avg} = V_{rms}I_{rms}\cos\phi$$
where $\cos\phi$ is called the power factor. For a pure resistor, $\phi=0°$ and $\cos\phi=1$ (maximum power transfer); for a pure inductor or capacitor, $\phi=90°$ and $\cos\phi=0$ (no net power is dissipated over a full cycle — energy is alternately stored and returned, not consumed).
Solved Example
An AC circuit has an RMS voltage of 200 V and RMS current of 5 A, with a phase difference of 60° between them. Calculate the average power consumed.
$P_{avg} = V_{rms}I_{rms}\cos\phi = (200)(5)\cos60° = (1000)(0.5) = 500\ \text{W}$
Q. Why does a pure inductor (or pure capacitor) consume zero net power on average in an AC circuit, despite having current flowing through it?
Ans: In a pure inductor or capacitor, voltage and current are 90° out of phase — during part of each cycle, the element absorbs energy (storing it in its magnetic or electric field), and during another part of the same cycle, it returns this exact same energy back to the circuit as the field collapses. Since energy absorbed equals energy returned within each cycle, the net energy consumed (and hence average power, since cosφ = cos90° = 0) is exactly zero, even though current is very much flowing through the element throughout.
Transformer
Principle
A transformer uses mutual induction between two coils (primary and secondary), wound on a common iron core, to step voltage up or down — it works only with AC (since a changing flux, and hence induced EMF, requires a continuously changing current).
Solved Example
A transformer has 500 turns on the primary and 2000 turns on the secondary. If the primary voltage is 220 V, calculate the secondary voltage.
$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} \Rightarrow V_s = V_p \times \dfrac{N_s}{N_p} = 220 \times \dfrac{2000}{500} = 220 \times 4 = 880\ \text{V}$
Q. Why can a transformer not be used to step up or step down a DC (direct current) voltage?
Ans: A transformer works through mutual induction, which requires a continuously changing magnetic flux in the primary coil to induce an EMF in the secondary coil. A steady DC current produces a constant, unchanging flux (after any brief initial transient), meaning there's no ongoing rate of change of flux to induce a continuous EMF in the secondary — so a transformer simply cannot step up or down a genuinely constant DC voltage in steady-state operation.
Q. A step-down transformer converts 440 V to 220 V. If the primary current is 2 A, calculate the secondary current (assuming an ideal, lossless transformer).
Ans: For an ideal transformer, input power = output power: VpIp = VsIs → (440)(2) = (220)Is → Is = 880/220 = 4 A.
Energy Losses in Real Transformers
Real transformers are never perfectly efficient, experiencing losses due to: copper loss (resistive heating, I²R, in the winding wires), iron/core loss (from both eddy currents and hysteresis in the magnetic core), and flux leakage (not all flux from the primary links perfectly with the secondary coil).
Solved Example
A transformer has an efficiency of 90%. If the input power is 1000 W, calculate the output power delivered.
$\text{Output power} = \text{Efficiency} \times \text{Input power} = 0.90 \times 1000 = 900\ \text{W}$
Q. Why is copper loss in a transformer's windings unavoidable, even in principle, unlike some other transformer losses that can be minimised through better design?
Ans: Copper loss arises directly from the resistive heating (I²R) inherent in any real wire carrying current, since no practical conducting wire has exactly zero resistance — while good conductor choice and thick, low-resistance windings can minimise this loss, it can never be reduced to absolute zero for any realistic transformer using ordinary conducting materials, unlike, say, eddy current losses, which can be dramatically (though still not perfectly) reduced through lamination of the core.
⚡ Formula Cheat-Sheet
Faraday's lawε = −N(dΦ/dt)
Motional EMFε = Bvl
Self-inductance (solenoid)L = μ₀N²A/l
Mutual EMFε₂ = −M(dI₁/dt)
AC generator EMFε = NBAω sin(ωt)
Inductive reactanceX_L = ωL
Capacitive reactanceX_C = 1/ωC
LCR impedanceZ = √(R² + (X_L−X_C)²)
Resonant frequencyω₀ = 1/√(LC)
Transformer equationVs/Vp = Ns/Np = Ip/Is