Unit 11: EMI, AC & EM Waves

EMI + AC + EM Waves carries 8–10 % of JEE-Main and is a stronghold of Advanced. The topic is unified by Faraday's law: a changing magnetic flux induces an EMF. From this single statement you derive motional EMF, self-induction, mutual induction, LC oscillations, AC circuits, transformer action, and (with Maxwell's correction) the entire theory of light.

Expect:

• JEE-Main: One motional-EMF problem (rod, disc, rotating ring), one self-inductance / mutual inductance / LR transient, one AC LCR analysis (impedance, phase, power, resonance), one EM-waves conceptual or numerical.
• JEE-Advanced: Combined problems — e.g. a rod sliding on rails forms an LR-EMF circuit; or a capacitor with changing field has a displacement current giving rise to a magnetic field, asked numerically.

The cleanest mental model:

• $\varepsilon = -d\Phi/dt$ (Faraday).
• AC steady state is best handled with impedances $Z_L = j\omega L, Z_C = 1/(j\omega C), Z_R = R$, then Ohm's law in phasor form.
• EM waves in vacuum: $\vec E, \vec B$ perpendicular to each other and to $\vec k$; $c = 1/\sqrt{\mu_0\varepsilon_0}$.

Concept Map

EMI
│
├── Flux Φ = ∫B·dA
├── Faraday:  ε = -dΦ/dt
├── Lenz: induced current opposes the change
├── Motional EMF:  ε = ∫(v×B)·dℓ
│     • rod sliding: ε = BvL
│     • rotating rod: ε = ½BωL²
│     • rotating disc: ε = ½BωR²
├── Self-inductance L:  ε_L = -L dI/dt
│     • Solenoid: L = μ₀n²V
├── Mutual inductance M:  ε₂ = -M dI₁/dt
│     • Reciprocity: M₁₂ = M₂₁
├── LR transient (growth/decay), τ = L/R
├── Energy in L: ½LI², density B²/(2μ₀)
└── LC oscillations:  ω = 1/√(LC)

AC CIRCUITS
│
├── v(t) = V₀ sin(ωt)
├── RMS:  V_rms = V₀/√2,  I_rms = I₀/√2
│
├── Pure elements
│     ├── R: i in phase with v
│     ├── L: i lags v by π/2
│     └── C: i leads v by π/2
│
├── Series LCR
│     Z = √(R² + (X_L − X_C)²),  X_L = ωL, X_C = 1/ωC
│     tan φ = (X_L − X_C)/R
│     Resonance: ω₀ = 1/√(LC), Z = R
│     Q = ω₀L/R = (1/R)√(L/C)
│
├── Power
│     P_avg = V_rms I_rms cos φ
│     Wattless: φ = π/2
│
└── Transformer
       V₂/V₁ = N₂/N₁ = I₁/I₂ (ideal)
       Efficiency, losses (Cu, hysteresis, eddy)

EM WAVES
│
├── Displacement current J_d = ε₀ dE/dt
├── Maxwell's equations (4)
├── c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s
├── E ⊥ B ⊥ k, E/B = c
├── Energy density u = ½ε₀E² + B²/(2μ₀); both equal
├── Intensity I = ½ε₀E₀²c = avg Poynting
├── Radiation pressure p = I/c (absorbed), 2I/c (reflected)
└── Spectrum: γ → X → UV → vis → IR → μw → radio

Topic 1: Electromagnetic Induction (Faraday & Lenz)

Sub-topic A: Magnetic Flux

For a flat coil in $\vec B$:

For non-uniform $B$ or curved surface:

Unit: weber (Wb) = T·m² = V·s.

Sub-topic B: Faraday's Law

The induced EMF in a closed loop equals the negative rate of change of magnetic flux through it. The sign embodies Lenz's law.

If the loop has resistance $R$, induced current $I = \varepsilon/R$.

Sub-topic C: Lenz's Law

The induced current flows in such a direction that its own magnetic field opposes the change in flux that produced it. This is the conservation of energy in induction — if it were the other way, you would have run-away currents (perpetual motion).

Practical use: sketch the original flux direction; sketch how it's changing; the induced current opposes that change.

Sub-topic D: Motional EMF — Three Standard Cases

Case 1: Straight rod sliding on rails.

Rod of length $L$ moves with velocity $v$ perpendicular to its length, in field $\vec B$ perpendicular to the plane.

Force on a free charge in the rod: $q\vec v\times\vec B$, which separates positive charge to one end (creates an electric field). At equilibrium $qE = qvB$, so the EMF (PD across the rod) is:

Equivalently, by Faraday: $\Phi = BL x$, $d\Phi/dt = BLv = \varepsilon$.

Case 2: Rotating rod.

Rod of length $L$ rotates with angular velocity $\omega$ about one end, in field $B$ perpendicular to the plane of rotation.

Element $d\xi$ at distance $\xi$ from the pivot moves with speed $\omega\xi$. EMF in element: $d\varepsilon = B(\omega\xi)d\xi$. Integrate:

Case 3: Rotating disc (Faraday disc).

A conducting disc of radius $R$ rotates at $\omega$ in field $B$ perpendicular to the disc. EMF between centre and rim is the integral of $d\varepsilon = B(\omega r)dr$ from $0$ to $R$:

Sub-topic E: Induced Electric Field

A changing magnetic field induces a non-conservative electric field even in the absence of charges. From Faraday in integral form:

Unlike the static electric field, this $\vec E$ has non-zero curl and its line integral around a closed path is non-zero.

For an axially symmetric region of uniform $B(t)$ in a circle of radius $R$:

Worked Problem 1

A horizontal conducting rod of length $1$ m falls freely under gravity in a horizontal magnetic field of $0.5$ T, with the rod perpendicular to $\vec B$. Find the EMF after $2$ s.

Solution. $v = gt = 20$ m/s. $\varepsilon = BvL = 0.5\cdot 20\cdot 1 = 10$ V.

Worked Problem 2 (JEE-Advanced)

A square loop of side $a$ and resistance $R$ moves with constant velocity $v$ into a region of uniform $B$ perpendicular to the loop's plane. The leading edge enters the field at $t = 0$. Find: (a) induced current while entering, (b) force needed to keep it moving, (c) power dissipated.

Solution. (a) While entering, only the leading edge is in the field; $\varepsilon = Bav$, $I = Bav/R$. (b) Force on the leading edge $F_{\text{mag}} = BIa = B^2a^2v/R$ (opposing motion). External force needed: $F_{\text{ext}} = B^2a^2v/R$. (c) Power dissipated: $P = I^2 R = (Bav)^2/R$. Equals $F_{\text{ext}}\cdot v$ — energy balance.

Once fully inside (or fully outside) $\Phi$ no longer changes, $I = 0$, no force needed.

Topic 2: Inductance

Sub-topic A: Self-Inductance

A current $I$ through a coil produces a flux $\Phi = LI$, where $L$ is the self-inductance (henry: H = Wb/A = V·s/A).

When $I$ changes:

This "back EMF" opposes the change in $I$.

Sub-topic B: Self-Inductance of a Long Solenoid

$n$ turns per unit length, area $A$, length $\ell$. Flux per turn $\Phi_1 = \mu_0 n I\cdot A$. Total flux linkage $N\Phi_1 = (n\ell)\mu_0 n I A = \mu_0 n^2 \ell A\cdot I$. So

where $V$ is the volume.

For a solenoid filled with a material of relative permeability $\mu_r$: $L \to \mu_0\mu_r n^2 V$.

Sub-topic C: Self-Inductance of a Toroid

Toroid of $N$ turns, mean radius $r$, cross-section $A$:

Sub-topic D: Mutual Inductance

If coil 1 has $I_1$ and produces flux $\Phi_{21}$ through coil 2:

$M$ is the mutual inductance; symmetric: $M_{12} = M_{21} = M$ (Reciprocity Theorem).

Two coaxial solenoids: small one (length $\ell$, area $A$, $n_1$ turns/m) inside a larger one ($n_2$ turns/m).

When current $I_2$ flows in the outer solenoid, field inside is $B = \mu_0 n_2 I_2$. Flux through the inner solenoid per turn: $BA = \mu_0 n_2 I_2 A$. Total flux linkage with the inner solenoid: $n_1\ell \cdot\mu_0 n_2 I_2 A$. So

Also $M = k\sqrt{L_1 L_2}$ where $0 \le k \le 1$ is the coupling coefficient. $k = 1$ for perfect (no leakage) flux linkage.

Sub-topic E: LR Circuit — Growth and Decay

Growth (switch closed at $t=0$, battery $\varepsilon$, resistor $R$, inductor $L$):

Solving:

Decay (battery removed at $t=0$, replaced by short):

Sub-topic F: Energy in an Inductor

Work done by EMF against back-EMF while current builds from 0 to $I$:

Energy density in the magnetic field:

This is the magnetic analog of $u_E = \tfrac{1}{2}\varepsilon_0 E^2$.

Sub-topic G: LC Oscillations — Derivation

Charged capacitor $C$ connected to inductor $L$. KVL:

SHM in charge! Angular frequency:

Total energy oscillates between capacitor ($q^2/2C$) and inductor ($\tfrac{1}{2}Li^2$), with sum constant:

Worked Problem 3

A $5$ mH inductor and a $20\,\mu$F capacitor are connected. Find the frequency of LC oscillations.

Solution. $\omega = 1/\sqrt{LC} = 1/\sqrt{5\times 10^{-3}\cdot 20\times 10^{-6}} = 1/\sqrt{10^{-7}} = 10^{3.5} = 3162$ rad/s. $f = 503$ Hz.

Topic 3: Alternating Current — Basics

Sub-topic A: Sinusoidal Source

Average over a cycle: zero (positive and negative halves cancel).

RMS (root mean square):

Same for $I_{\text{rms}} = I_0/\sqrt 2$.

Power: with resistive load $\langle P\rangle = V_{\text{rms}} I_{\text{rms}} = V_0^2/(2R)$.

Sub-topic B: Pure Elements with AC

Pure resistor:

$i$ in phase with $v$.

Pure inductor:

Current lags voltage by $\pi/2$. Inductive reactance $X_L = \omega L$.

Pure capacitor:

Current leads voltage by $\pi/2$. Capacitive reactance $X_C = 1/(\omega C)$.

Sub-topic C: Phasor Representation

Visualize AC quantities as rotating vectors (phasors) — projection on a chosen axis gives the instantaneous value. Adding two AC quantities of the same frequency reduces to phasor addition.

In phasor diagrams (with current as reference):

• Resistor voltage: along $\hat I$.
• Inductor voltage: 90° ahead of $\hat I$.
• Capacitor voltage: 90° behind $\hat I$.

Topic 4: Series LCR Circuit

Sub-topic A: Impedance and Phase

Series LCR with sinusoidal source. KVL (phasor):

Magnitudes: $V_R = IR$, $V_L = IX_L$, $V_C = IX_C$. The $V_L$ and $V_C$ are antiparallel; the net reactive voltage is $V_L - V_C$ (if $V_L > V_C$).

Magnitude of total voltage:

where

Phase angle of voltage w.r.t. current:

If $\phi > 0$, voltage leads current (inductive); if $\phi < 0$, voltage lags (capacitive).

Sub-topic B: Resonance

Resonance at $X_L = X_C \Rightarrow \omega_0 = 1/\sqrt{LC}$, the same frequency as LC oscillations.

At resonance:

• $Z = R$ (minimum).
• $I$ is maximum: $I_0 = V_0/R$.
• $\phi = 0$: current in phase with applied voltage.
• $V_L$ and $V_C$ individually can be much larger than the source voltage but cancel each other out.

Bandwidth $\Delta\omega = R/L$. Quality factor:

Higher $Q$ = sharper resonance.

Sub-topic C: Power and Power Factor

Instantaneous power $p(t) = v(t)\cdot i(t)$. Average over a cycle:

$\cos\phi$ is the power factor. Pure inductor or capacitor has $\phi = \pm\pi/2$, $\cos\phi = 0$: no average power dissipated — "wattless current".

In LCR: only $R$ dissipates power, and that's $\langle P\rangle = I_{\text{rms}}^2 R$.

Worked Problem 4

A series LCR has $R = 100\,\Omega$, $L = 0.5$ H, $C = 10\,\mu$F. Source $V_0 = 100$ V, frequency $50$ Hz. Find $X_L, X_C, Z, I_0$, phase, average power.

Solution. $\omega = 100\pi \approx 314$ rad/s.

$X_L = \omega L = 157\,\Omega$.

$X_C = 1/(\omega C) = 1/(314\cdot 10^{-5}) = 318\,\Omega$.

$X_L - X_C = -161\,\Omega$ (capacitive).

$Z = \sqrt{100^2 + 161^2} = \sqrt{10000+25921}\approx 189.5\,\Omega$.

$I_0 = 100/189.5 \approx 0.528$ A.

$\tan\phi = -161/100 = -1.61 \Rightarrow \phi \approx -58.2^\circ$ (voltage lags current).

$\langle P\rangle = V_{\text{rms}}I_{\text{rms}}\cos\phi = (100/\sqrt 2)(0.528/\sqrt 2)(0.527) \approx 13.9$ W.

Worked Problem 5 (Resonance)

For the same LCR, find the resonance frequency.

Solution. $\omega_0 = 1/\sqrt{LC} = 1/\sqrt{0.5\cdot 10^{-5}} = 1/\sqrt{5\times 10^{-6}}\approx 447$ rad/s, $f_0 \approx 71.2$ Hz.

Topic 5: Transformers

Sub-topic A: Principle

A transformer has two coils wound on the same iron core. AC current in the primary creates a changing flux; the secondary picks up the flux through mutual induction. With $N_1, N_2$ turns, for an ideal transformer (no losses):

Power conservation (ideal): $V_1 I_1 = V_2 I_2$ ⇒

Step-up: $N_2 > N_1$ ⇒ $V_2 > V_1$ (and $I_2 < I_1$). Step-down: $N_2 < N_1$ ⇒ $V_2 < V_1$ (and $I_2 > I_1$).

Sub-topic B: Losses

1. Copper loss ($I^2R$ in the windings).
2. Iron / hysteresis loss (energy in magnetisation cycle — minimised by soft iron).
3. Eddy current loss (induced currents in the core — minimised by laminated cores).
4. Flux leakage (not all flux from primary reaches secondary).

Real transformer efficiency $\eta = P_{\text{out}}/P_{\text{in}}\approx 95$–$99\%$.

Transformers don't work on DC (no $d\Phi/dt$).

Topic 6: Maxwell's Equations & Electromagnetic Waves

Sub-topic A: Displacement Current

Maxwell noticed: Ampère's law fails for a capacitor being charged. The "current" enclosed depends on which surface you take spanning a given loop! He fixed it by introducing the displacement current:

Modified Ampère–Maxwell law:

Sub-topic B: Maxwell's Equations (Integral Form)

Equation #4's displacement current is what closes the loop and predicts EM waves.

Sub-topic C: EM Waves in Vacuum

Combining Maxwell's equations in free space gives:

These are wave equations with speed

Plugging numbers: $c = 1/\sqrt{4\pi\times 10^{-7}\cdot 8.854\times 10^{-12}} \approx 3\times 10^8$ m/s — exactly the speed of light. Light is an EM wave.

Properties of EM waves in vacuum:

1. Transverse: $\vec E, \vec B$ both perpendicular to direction of propagation $\hat k$.
2. $\vec E$ and $\vec B$ are perpendicular to each other.
3. In phase (peak together, zero together).
4. $E_0 = c B_0 \Leftrightarrow E/B = c$.
5. Direction of propagation $\hat k = \hat E\times\hat B$.

Sub-topic D: Energy and Momentum of EM Waves

Energy densities (in vacuum, instantaneous):

Using $E = cB$ and $c^2 = 1/(\mu_0\varepsilon_0)$:

Equal! Total energy density:

Time-averaged for sinusoidal wave ($\langle E^2\rangle = E_0^2/2$):

Intensity $I$ (avg energy crossing unit area per unit time):

Poynting vector: $\vec S = (1/\mu_0)\vec E\times\vec B$ (instantaneous energy flux).

Radiation pressure on a perfectly absorbing surface:

For a perfectly reflecting surface (momentum reversal):

Momentum density in field: $u/c = \varepsilon_0 E^2/c$.

Sub-topic E: EM Spectrum

All travel at $c$ in vacuum; ordering is by frequency (γ highest, radio lowest).

Worked Problem 6

A 100 W lamp radiates uniformly in all directions. At distance $1$ m, find $E_0$, $B_0$, intensity, and radiation pressure on a perfect absorber.

Solution. $I = P/(4\pi r^2) = 100/(4\pi) \approx 7.96$ W/m².

$E_0 = \sqrt{2I/(\varepsilon_0 c)} = \sqrt{2\cdot 7.96/(8.85\times 10^{-12}\cdot 3\times 10^8)} = \sqrt{5994}\approx 77.4$ V/m.

$B_0 = E_0/c = 2.58\times 10^{-7}$ T.

Radiation pressure $P = I/c = 7.96/(3\times 10^8) = 2.65\times 10^{-8}$ Pa. Tiny — but used by solar sails.

Problem-Solving Heuristics

1. For motional EMF, use $\varepsilon = \int(\vec v\times\vec B)\cdot d\vec\ell$ rather than trying to find $d\Phi/dt$ for rotating systems.
2. For an LR transient, $\tau = L/R$; for RC, $\tau = RC$. After $5\tau$, you're at steady state.
3. At steady state in DC, inductors act as wires ($V_L = L\,dI/dt = 0$) and capacitors as open circuits ($I_C = 0$).
4. At $t = 0^+$ (just after a switch), inductors prevent sudden change in current (act as open), capacitors prevent sudden change in voltage (act as short).
5. Phasors: for an LCR problem, always draw the phasor diagram with $\vec I$ as reference, then $V_R$ along it, $V_L$ up, $V_C$ down, and total $V$ as the resultant.
6. At resonance $X_L = X_C$, $Z = R$ (smallest possible), $I$ maximum.
7. Power factor: $\cos\phi = R/Z$. Wattless if $\phi = \pm\pi/2$ (pure $L$ or $C$).
8. For transformer problems, conserve power if ideal; for non-ideal, $\eta = V_2I_2\cos\phi_2/(V_1I_1\cos\phi_1)$.
9. In EM waves, $\vec k\propto \vec E\times\vec B$ (right-hand rule). $E_0 = cB_0$.
10. Energy density $u = \varepsilon_0 E^2$ total in vacuum (electric and magnetic equal).
11. For reflected light, radiation pressure is twice the absorber case ($2I/c$).
12. Identify the loop / surface for Faraday consistently — choose normals; sign conventions matter.

Common Traps & Mistakes

• Lenz's law sign. The induced EMF opposes the change in flux, not the flux itself.
• Self-inductance is always positive, but the EMF is opposite to $dI/dt$.
• In an LCR at resonance, $V_L$ and $V_C$ can each exceed source $V$, but they cancel — so $V$ across $L+C$ = 0 only when summed as phasors.
• Reactance $X_L$ and $X_C$ depend on $\omega$ — don't use static values.
• Average power across a pure $L$ or $C$ is zero, not just for one quarter cycle.
• Transformers don't increase or decrease power (ideal), only redistribute V and I.
• Maxwell's correction (displacement current) isn't real charge motion — it's $\varepsilon_0\,dE/dt$.
• EM waves carry momentum: $p = U/c$. Most students forget this.
• For EM waves in a medium, $v = c/n$, $\lambda_n = \lambda_0/n$, but frequency is unchanged.
• A current in DC doesn't see inductance — only when switching. At steady DC, inductors are wires.
• Energy in an inductor is $\tfrac{1}{2}LI^2$, not $LI$.

Quick Revision Card

• Flux: $\Phi = NBA\cos\theta$.
• Faraday: $\varepsilon = -d\Phi/dt$; Lenz says induced opposes the change.
• Motional EMF: rod $\varepsilon = BvL$; rotating rod $\tfrac{1}{2}B\omega L^2$; disc $\tfrac{1}{2}B\omega R^2$.
• Self-inductance: $L = \mu_0 n^2 V$ (solenoid).
• Mutual inductance: $M = \mu_0 n_1 n_2\ell A$, $M = k\sqrt{L_1L_2}$.
• LR: $I = (\varepsilon/R)(1 - e^{-t/\tau})$, $\tau = L/R$. Energy $\tfrac{1}{2}LI^2$, density $B^2/(2\mu_0)$.
• LC: $\omega = 1/\sqrt{LC}$.
• AC: $V_{rms} = V_0/\sqrt 2$.
• Pure $L$: $i$ lags $v$ by $\pi/2$, $X_L = \omega L$.
• Pure $C$: $i$ leads $v$ by $\pi/2$, $X_C = 1/(\omega C)$.
• Series LCR: $Z = \sqrt{R^2 + (X_L - X_C)^2}$, $\tan\phi = (X_L-X_C)/R$.
• Resonance: $\omega_0 = 1/\sqrt{LC}$, $Z = R$, $Q = (1/R)\sqrt{L/C}$.
• Power: $\langle P\rangle = V_{rms}I_{rms}\cos\phi$.
• Transformer: $V_2/V_1 = N_2/N_1 = I_1/I_2$.
• Displacement current: $I_d = \varepsilon_0\,d\Phi_E/dt$.
• EM wave: $c = 1/\sqrt{\mu_0\varepsilon_0}$; $E = cB$; $u = \varepsilon_0 E^2$; $I = \tfrac{1}{2}\varepsilon_0 E_0^2 c$.
• Radiation pressure: $I/c$ (absorbed), $2I/c$ (reflected).

Formula Sheet