Electromagnetic Induction and Alternating Currents — Practice Questions with Answers

First law: Whenever the magnetic flux linked with a circuit changes, an EMF is induced in the circuit, lasting as long as the flux continues to change.

Second law: The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linkage: $\varepsilon = -\dfrac{d\Phi}{dt}$ (for a single-turn coil), or $\varepsilon = -N\dfrac{d\Phi}{dt}$ for $N$ turns.

Lenz's law states that the direction of an induced current is always such that it opposes the change in magnetic flux that produced it.

Significance: This opposing nature ensures energy conservation — if the induced current instead aided the change (rather than opposing it), it would create a self-reinforcing process generating energy from nothing, violating energy conservation. Since the induced current opposes the change, external work must be done to maintain or continue that change, and this external work is precisely what supplies the electrical energy manifested as the induced current.

Self-inductance is the property of a coil by which it opposes any change in the current flowing through itself, by inducing an EMF within the coil due to the changing magnetic flux this changing current itself creates.

Since flux linkage $N\Phi \propto I$, we can write $N\Phi = LI$, where $L$ is the self-inductance. Since induced EMF $\varepsilon = -\dfrac{d(N\Phi)}{dt}$:

$\varepsilon = -L\dfrac{dI}{dt}$

An AC generator converts mechanical energy into electrical energy, based on electromagnetic induction. A coil is mechanically rotated within a uniform magnetic field (or, equivalently, a magnet is rotated near a stationary coil) — as the coil rotates, the magnetic flux linked with it continuously changes.

This continuously changing flux induces an alternating EMF in the coil (following Faraday's law), given by $\varepsilon = NBA\omega\sin(\omega t)$ — a sinusoidally varying EMF, which is the defining characteristic of alternating current.

The RMS (root-mean-square) value of an alternating current is the value of steady direct current that would produce the same amount of heat (same power dissipation) in a given resistance over one complete cycle as the actual alternating current does.

It is more practically useful than peak value because AC power dissipation ($I^2R$) directly depends on RMS current, not peak current — quoting standard household voltage (e.g., 220V AC) as an RMS value directly and meaningfully indicates its actual heating/power delivery capability, unlike the instantaneously varying peak value.

An inductor's opposing (back) EMF is given by $\varepsilon = -L\dfrac{dI}{dt}$ — this depends specifically on the rate of change of current, not on the current itself.

For a steady DC current, $I$ is constant, so $\dfrac{dI}{dt} = 0$, meaning the inductor generates no back-EMF and offers no additional opposition beyond its own small wire resistance. Only when current is changing (as during switching transients, or continuously in AC) does the inductor generate a genuine opposing EMF, resisting that specific change.

Impedance is the total effective opposition an AC circuit offers to current flow, combining resistance and reactance (from inductors and capacitors).

For a series LCR circuit: $Z = \sqrt{R^2 + (X_L-X_C)^2}$, where $X_L = \omega L$ (inductive reactance) and $X_C = \dfrac{1}{\omega C}$ (capacitive reactance) — since inductive and capacitive reactances have opposite phase effects, they partially (or fully, at resonance) cancel within the impedance formula.

Resonance occurs in a series LCR circuit when the inductive reactance exactly equals the capacitive reactance ($X_L=X_C$), causing these opposing reactive effects to cancel completely, leaving impedance at its minimum value ($Z=R$) and current at its maximum value for a given applied voltage.

Setting $X_L=X_C$: $\omega L = \dfrac{1}{\omega C} \Rightarrow \omega^2 = \dfrac{1}{LC} \Rightarrow \omega_0 = \dfrac{1}{\sqrt{LC}}$

Resonant frequency: $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$

A solid metal core would allow large eddy currents to circulate within its bulk due to the continuously changing magnetic flux (from the AC input), causing significant energy loss as heat (and reducing transformer efficiency).

By laminating the core — using thin sheets electrically insulated from each other, rather than one solid block — the pathways available for eddy currents to circulate are broken up and significantly restricted, substantially reducing eddy current losses and improving the transformer's overall efficiency.

A step-up transformer increases voltage from the primary (input) coil to the secondary (output) coil. Since the same changing flux links both coils (via a shared iron core), the induced EMF in each coil is proportional to its number of turns: $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$.

By constructing the secondary coil with more turns than the primary ($N_s > N_p$), the output voltage $V_s$ becomes correspondingly larger than the input voltage $V_p$ — this is precisely the working principle of a step-up transformer, achieved purely through the turns ratio, based on mutual induction.

$\varepsilon = N\dfrac{\Delta\Phi}{\Delta t} = 200\times\dfrac{5\times10^{-3}-0}{0.1}$

$\varepsilon = 200\times\dfrac{5\times10^{-3}}{0.1} = 200\times0.05$

$\varepsilon = \mathbf{10\text{ V}}$

Secondary voltage: $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} \Rightarrow V_s = 220\times\dfrac{100}{1000} = 22\text{ V}$

Primary current (for an ideal transformer, power in = power out): $V_pI_p = V_sI_s \Rightarrow I_p = \dfrac{V_sI_s}{V_p} = \dfrac{22\times5}{220}$

$I_p = \dfrac{110}{220} = \mathbf{0.5\text{ A}}$

$V_{rms} = \dfrac{V_0}{\sqrt{2}} = \dfrac{311}{1.414}$

$V_{rms} \approx \mathbf{220\text{ V}}$

This confirms the standard 220 V household AC supply corresponds to a peak voltage of about 311 V.

(a) Impedance: $Z = \sqrt{R^2+(X_L-X_C)^2} = \sqrt{30^2+(50-10)^2} = \sqrt{900+1600} = \sqrt{2500} = 50\,\Omega$

(b) Current: $I = \dfrac{V}{Z} = \dfrac{200}{50} = \mathbf{4\text{ A}}$

$f_0 = \dfrac{1}{2\pi\sqrt{LC}}$

$LC = 0.5\times8\times10^{-6} = 4\times10^{-6}$

$\sqrt{LC} = \sqrt{4\times10^{-6}} = 2\times10^{-3}$

$f_0 = \dfrac{1}{2\pi\times2\times10^{-3}} = \dfrac{1}{4\pi\times10^{-3}} \approx \dfrac{1}{0.01257}$

$f_0 \approx \mathbf{79.6\text{ Hz}}$

Motional EMF: $\varepsilon = Blv$

$\varepsilon = 0.8\times0.5\times4$

$\varepsilon = \mathbf{1.6\text{ V}}$

$\varepsilon = L\dfrac{\Delta I}{\Delta t} = 2\times\dfrac{|1-5|}{0.2} = 2\times\dfrac{4}{0.2}$

$\varepsilon = 2\times20 = \mathbf{40\text{ V}}$

$X_L = \omega L = 2\pi f L$

$X_L = 2\pi\times50\times0.2$

$X_L = 2\pi\times10 = 20\pi$

$X_L \approx \mathbf{62.8\,\Omega}$

$X_C = \dfrac{1}{\omega C} = \dfrac{1}{2\pi fC}$

$X_C = \dfrac{1}{2\pi\times60\times10\times10^{-6}}$

$X_C = \dfrac{1}{2\pi\times6\times10^{-4}} = \dfrac{1}{3.77\times10^{-3}}$

$X_C \approx \mathbf{265.3\,\Omega}$

Peak EMF for an AC generator: $\varepsilon_0 = NBA\omega$

$\varepsilon_0 = 100\times0.4\times0.02\times50$

$\varepsilon_0 = 100\times0.4\times1$

$\varepsilon_0 = \mathbf{40\text{ V}}$