EMI

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

EMI and AC — Key Facts

Electromagnetic Induction: Faraday’s First Law: When the magnetic flux through a closed circuit changes, an emf is induced in the circuit. The induced emf lasts only as long as the change in flux continues.

Faraday’s Second Law: The magnitude of induced emf is proportional to the rate of change of magnetic flux: $\varepsilon = -\frac{d\Phi}{dt}$. The minus sign encodes Lenz’s Law.

Lenz’s Law: The direction of induced current is such that it opposes the change in magnetic flux that caused it. This is a consequence of conservation of energy — the induced current does work against the change, and this work requires energy input.

Motional EMF: A rod of length $\ell$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$: $\varepsilon = B\ell v$. More generally: $\varepsilon = B\ell v\sin\theta$ where $\theta$ is the angle between $v$ and $B$. If the rod rotates about one end in a uniform $B$: $\varepsilon = \frac{1}{2}B\omega\ell^2 = \frac{1}{2}Bl^2\omega$.

Self-Induction: When current in a coil changes, the changing magnetic flux links back to the same coil, inducing an emf: $\varepsilon = -L\frac{dI}{dt}$. $L$ (self-inductance) depends on geometry:

• Solenoid: $L = \mu_0 N^2 A/\ell$
• Toroid: $L = \mu_0 N^2 r/(2\pi R)$ (where $r$ = cross-section radius, $R$ = mean radius)

Energy stored in an inductor: $U = \frac{1}{2}LI^2$. Energy density in a magnetic field: $u = \frac{B^2}{2\mu_0}$ (in vacuum).

⚡ Exam tip: Lenz’s Law always conserves energy — the induced current creates a magnetic field that opposes the original change. If the flux is increasing, the induced current flows to create a field in the opposite direction; if decreasing, it flows to reinforce the original field.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

EMI and AC — CUET Study Guide

Mutual Induction: When current $I_1$ in coil 1 changes, it induces an emf $\varepsilon_2$ in coil 2: $\varepsilon_2 = -M\frac{dI_1}{dt}$. The mutual inductance $M = k\sqrt{L_1 L_2}$ where $k$ is the coupling coefficient ($0 \leq k \leq 1$).

Transformer: An ideal transformer with $N_p$ primary turns and $N_s$ secondary turns: $V_s/V_p = N_s/N_p = I_p/I_s$. For a step-up transformer, $N_s > N_p$ and $V_s > V_p$ but $I_s < I_p$ (power $V \times I$ is conserved). Efficiency $\eta = (P_{out}/P_{in}) \times 100%$. Ideal transformer: $\eta = 100%$. Practical: $\eta ≈ 90-98%$ due to core losses (hysteresis + eddy currents).

Eddy current losses: Changing magnetic flux in a conductor induces circulating currents (eddy currents) that dissipate energy as heat. Reduced by laminating the core (thin insulated sheets) or using high-resistivity materials.

AC Generator: 原理: A rectangular coil rotating in a uniform magnetic field. The emf induced: $\varepsilon = \varepsilon_0 \sin(\omega t)$ where $\varepsilon_0 = NBA\omega$ is the peak emf. $N$ = number of turns, $B$ = magnetic field strength, $A$ = area of coil, $\omega$ = angular velocity.

RMS (root mean square) values: For sinusoidal AC, $I_{rms} = I_0/\sqrt{2}$, $V_{rms} = V_0/\sqrt{2}$. Mains voltage of 230 V AC means $V_{rms} = 230$ V, so $V_0 = 230\sqrt{2} ≈ 325$ V.

AC Through Individual Circuit Elements:

• Resistor: $V = IR$, current and voltage are in phase ($\phi = 0°$)
• Inductor: $V = I X_L$ where $X_L = \omega L = 2\pi fL$ (inductive reactance). Current lags voltage by $90°$ ($\phi = +90°$)
• Capacitor: $V = I X_C$ where $X_C = 1/(\omega C) = 1/(2\pi fC)$ (capacitive reactance). Current leads voltage by $90°$ ($\phi = -90°$)

LC Oscillations: An LC circuit (no resistance) oscillates with natural frequency $\omega_0 = 1/\sqrt{LC}$. The total energy $E = \frac{1}{2}CV^2 + \frac{1}{2}LI^2$ is constant, oscillating between the capacitor (electric field) and the inductor (magnetic field). Analogy: LC oscillation = spring-mass mechanical oscillator ($L \leftrightarrow m$, $C \leftrightarrow 1/k$).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

EMI and AC — Comprehensive Physics Notes

Faraday’s Law — Complete Derivation:

Consider a coil of area $A$ in a uniform magnetic field $B$ making an angle $\theta$ with the normal: $\Phi = NBA\cos\theta$. If $\theta$ changes with time, $\varepsilon = -d\Phi/dt = -NA \frac{d(B\cos\theta)}{dt}$. If $B$ changes: $\varepsilon = -NA\cos\theta \frac{dB}{dt}$. If both change: $\varepsilon = -NA(B\frac{d(\cos\theta)}{dt} + \cos\theta\frac{dB}{dt})$.

Growing and Decaying AC: For a sinusoidal emf applied to an RL circuit: $L\frac{dI}{dt} + RI = \varepsilon_0 \sin(\omega t)$. Solution has two parts: transient (decaying exponential) and steady state $I = I_0 \sin(\omega t - \phi)$ where $I_0 = \varepsilon_0/Z$, $Z = \sqrt{R^2 + (\omega L)^2}$, $\tan\phi = \omega L/R$.

Series LCR Circuit: Impedance $Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (\omega L - 1/(\omega C))^2}$. Phase angle $\tan\phi = (X_L - X_C)/R = (\omega L - 1/(\omega C))/R$.

At resonance ($\omega = \omega_r = 1/\sqrt{LC}$): $X_L = X_C$, $Z = R$ (minimum), $\phi = 0°$, current is maximum ($I_0 = V_0/R$), power dissipated is maximum.

Resonance in Series LCR: At resonance: $\omega_r = 1/\sqrt{LC}$. Quality factor $Q = \omega_r L/R = 1/(\omega_r CR)$. Bandwidth $\Delta\omega = R/L$. High $Q$ means a narrow resonance peak — the circuit is selective but easily disturbed. Low $Q$ means a broad peak but more stable.

Practical resonance curves: At $\omega = \omega_r$, current is maximum. At $\omega_1$ and $\omega_2$ (where $Z = \sqrt{2}R$), the power is half the maximum — these are the half-power points. $\Delta\omega = \omega_2 - \omega_1 = R/L$.

AC Power: Instantaneous power: $P(t) = V(t)I(t) = V_0 I_0 \sin\omega t \sin(\omega t - \phi)$. Average power: $\bar{P} = V_{rms} I_{rms} \cos\phi$. $\cos\phi$ is the power factor. For a purely resistive circuit ($\phi = 0$): $\bar{P} = V_{rms} I_{rms}$. For purely reactive ($\phi = 90°$): $\bar{P} = 0$ (no net power dissipation — energy flows back and forth between source and reactive elements).

For motors and transformers, a low power factor is inefficient (apparent power $VI$ is large relative to real power $VI\cos\phi$). Power factor correction uses capacitors to bring $\cos\phi$ closer to 1.

Dimensional Analysis of EMI Quantities:

• Magnetic flux $\Phi = B \cdot A$: $[Φ] = Wb = T·m² = (kg·s⁻²·A⁻¹)·m² = kg·m²·s⁻¹·A⁻¹$
• Inductance $L = \Phi/I$: $[L] = H = Wb/A = kg·m²·s⁻²·A⁻²$
• Reactance $X = V/I$: $[X] = Ω = kg·m²·s⁻³·A⁻²$

Transformer — Detailed Analysis: Ideal transformer (no losses): $P_{in} = P_{out}$. $V_p I_p = V_s I_s$. Efficiency $\eta = (V_s I_s)/(V_p I_p) \times 100%$. Practical transformer has losses from: copper resistance (I²R losses), hysteresis (domain wall movement in core), eddy currents (circulating currents in core), and flux leakage (not all flux links both coils).

Step-up: $N_s > N_p$, $V_s > V_p$, $I_s < I_p$. Step-down: opposite.

Induction Heating: eddy currents induced in a conductor by an alternating magnetic field dissipate as heat: $P = K I_p^2 R_{eddy} f^2$ where $f$ = frequency. Used in induction cooktops, metallurgical furnaces, and hyperthermia treatment.

AC vs DC: DC: constant direction and magnitude. AC: changes direction periodically. Advantage of AC: voltage can be changed easily using transformers (step up for transmission to reduce $I^2R$ losses, step down for consumer use). This is why mains electricity is AC at 50/60 Hz rather than DC.

JEE Pattern Analysis: CUET/JEE questions frequently ask: (1) Motional emf $B\ell v\sin\theta$, (2) Self-inductance of solenoid, (3) LC oscillations and $\omega_0 = 1/\sqrt{LC}$, (4) Power in AC $P = V_{rms} I_{rms} \cos\phi$, (5) Resonant frequency and Q-factor. JEE 2022: “A rod of length $\ell$ rotates about one end with angular velocity $\omega$ in a perpendicular magnetic field $B$. Find the emf induced between its ends.” Answer: $\varepsilon = \frac{1}{2}B\omega\ell^2$ (integrating $d\varepsilon = Bx\omega,dx$ from 0 to $\ell$).

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