Unit 11: Magnetism, EMI, AC and EM Waves

A heavyweight unit: 3–4 MCQs in NEET. It combines magnetism (Biot-Savart, Ampere's law, force on currents) with electromagnetic induction, AC circuits (LCR, resonance, transformer), and the brief topic of EM waves. Questions mix conceptual (rule applications, direction of induced current) with formula-recall (RMS values, resonance frequency, transformer ratio).

Concept Map

• Magnetic field of currents — Biot-Savart, Ampere's law (straight wire, loop, solenoid, toroid)
• Lorentz force, motion of charge in B, cyclotron
• Force on current-carrying wire, force between parallel wires, torque on loop, galvanometer (A, V)
• Magnetism and matter — dia/para/ferro, susceptibility, hysteresis
• EMI — Faraday, Lenz, motional EMF, self/mutual inductance
• AC — RMS, R/L/C alone, LCR, resonance, power factor, transformer
• EM waves — Maxwell brief, $c = 1/\sqrt{\mu_0\varepsilon_0}$, spectrum

Topic 1: Magnetic Field due to Currents

Sub-topic A: Biot-Savart Law

with $\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$.

Sub-topic B: Field of a Straight Wire

For a finite straight wire, field at perpendicular distance $a$:

For infinitely long wire: $\theta_1 = \theta_2 = 90°$,

Direction by right-hand rule (curl of fingers along $B$, thumb along $I$).

Sub-topic C: Field of a Circular Loop

At centre of a loop of radius $R$ carrying current $I$:

For $N$ turns: $B = \mu_0 N I/(2R)$.

On axis at distance $x$:

At large distance, $B \approx \mu_0 I R^2/(2 x^3) = \mu_0 m/(2\pi x^3)$ where $m = I A = I\pi R^2$ is the magnetic moment.

Sub-topic D: Ampere's Law

Use Amperian loops to find $B$ for symmetric currents:

• Long solenoid (n turns per unit length): $B = \mu_0 n I$ inside, $B \approx 0$ outside.
• Toroid of $N$ turns at mean radius $r$: $B = \mu_0 N I/(2\pi r)$.

Topic 2: Lorentz Force and Motion in B

Sub-topic A: Lorentz Force

The magnetic part does no work ($\vec F \perp \vec v$).

Sub-topic B: Charged Particle in Uniform B

If $\vec v \perp \vec B$: circular motion of radius

The period is independent of speed — basis of cyclotron.

If $\vec v$ has component along $\vec B$: helical motion. Pitch $p = v_\parallel T$.

Sub-topic C: Cyclotron

Particle accelerated by alternating voltage between dees, magnetically curved. Cyclotron frequency $f_c = qB/(2\pi m)$. Max KE:

Relativistic limit: at high speed mass increases, cyclotron period changes — formula breaks down.

Topic 3: Force on Currents

Sub-topic A: Force on a Current Wire

Sub-topic B: Two Parallel Wires

Force per unit length between two long parallel wires carrying $I_1, I_2$, separated by $d$:

Attractive if currents are parallel; repulsive if antiparallel. (This defines the ampere.)

Sub-topic C: Torque on a Current Loop

A loop of $N$ turns, area $A$, carrying current $I$ in field $B$ has magnetic moment $m = NIA$. Torque:

PE: $U = -\vec m \cdot \vec B$.

Sub-topic D: Moving-Coil Galvanometer

Deflection $\phi = (NAB/k)I$, where $k$ is restoring torsion constant. Current sensitivity: $\phi/I = NAB/k$. Voltage sensitivity: $\phi/V = NAB/(kR_g)$.

Ammeter: galvanometer in parallel with a low shunt $S = G I_g/(I - I_g)$.

Voltmeter: galvanometer in series with a high $R = V/I_g - G$.

Topic 4: Magnetism and Matter

Sub-topic A: Magnetic Susceptibility and Permeability

Relative permeability $\mu_r = 1 + \chi$.

Sub-topic B: Types of Materials

Above the Curie temperature, ferromagnet becomes paramagnet.

Sub-topic C: Hysteresis

The $B$-$H$ curve for a ferromagnet is a closed loop. Key points: residual magnetism (retentivity), coercivity (the $H$ to demagnetize). Area = energy dissipated per cycle.

• Soft iron: low retentivity, low coercivity → electromagnets, transformer cores.
• Steel: high retentivity, high coercivity → permanent magnets.

Topic 5: Electromagnetic Induction

Sub-topic A: Faraday's Law

EMF induced in a closed loop = rate of change of flux:

For $N$ turns: $\varepsilon = -N\,d\Phi_B/dt$.

Sub-topic B: Lenz's Law

The direction of the induced current is such that it opposes the change that produces it (conservation of energy).

Sub-topic C: Motional EMF

A rod of length $L$ moving with velocity $v$ in field $B$ (all mutually perpendicular):

If the rod is part of a circuit of resistance $R$: induced current $I = BLv/R$; retarding force $F = BIL = B^2 L^2 v/R$; power dissipated $P = B^2L^2v^2/R = Fv$.

Sub-topic D: Self-Inductance

For long solenoid: $L = \mu_0 n^2 A \ell$.

Energy stored: $U = \tfrac{1}{2} L I^2$. Energy density: $u = B^2/(2\mu_0)$.

Sub-topic E: Mutual Inductance

For two coaxial solenoids: $M = \mu_0 n_1 n_2 A \ell$.

Topic 6: Alternating Current

Sub-topic A: AC Voltage and Current

$V(t) = V_0\sin\omega t$, $I(t) = I_0\sin(\omega t - \phi)$.

• Peak value: $V_0$.
• RMS value: $V_\text{rms} = V_0/\sqrt{2}$. Similarly $I_\text{rms} = I_0/\sqrt{2}$.
• Average over half cycle: $V_\text{avg} = 2V_0/\pi$.

Sub-topic B: Single-Element Circuits

Sub-topic C: Series LCR Circuit

Impedance:

Current amplitude $I_0 = V_0/Z$.

Phase $\tan\phi = (X_L - X_C)/R$.

Average power:

$\cos\phi$ is the power factor. For pure L or pure C, $\cos\phi = 0$ → no average power dissipated (wattless current).

Sub-topic D: Resonance

Series LCR resonates when $X_L = X_C$:

At resonance: $Z = R$ (minimum), $I_0 = V_0/R$ (maximum), $\cos\phi = 1$.

Quality factor:

High Q → sharp resonance, narrow bandwidth.

Sub-topic E: Transformer

Ideal transformer: $V_p/V_s = N_p/N_s$ and $I_p/I_s = N_s/N_p$. Step-up: $N_s > N_p$, $V_s > V_p$ but $I_s < I_p$.

Efficiency $\eta = P_s/P_p$. Losses: copper (I²R), iron (hysteresis + eddy current), flux leakage.

Topic 7: Electromagnetic Waves

Sub-topic A: Displacement Current

Maxwell added the displacement current $I_d = \varepsilon_0\,d\Phi_E/dt$ to make Ampere's law consistent. So

Sub-topic B: Maxwell's Equations (qualitative)

Four equations: Gauss for E, Gauss for B (no monopoles), Faraday's law, Ampere-Maxwell. They predict EM waves.

Sub-topic C: Properties of EM Waves

• Transverse: $\vec E \perp \vec B \perp \vec k$ (propagation direction).
• Speed in vacuum: $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3 \times 10^8$ m/s.
• In a medium: $v = 1/\sqrt{\mu\varepsilon} = c/n$.
• $E_0/B_0 = c$.
• Energy density: $u = \tfrac{1}{2}\varepsilon_0 E^2 + B^2/(2\mu_0)$. For sinusoidal wave, average $u = \tfrac{1}{2}\varepsilon_0 E_0^2$.
• Intensity (Poynting): $S = E B/\mu_0 = u c$.
• Carry momentum: $p = E/c$ per photon; produce radiation pressure $P = I/c$ (absorbed) or $2I/c$ (reflected).

Sub-topic D: EM Spectrum (Memorise order)

Visible (mnemonic VIBGYOR): violet ($\sim$ 400 nm) → red ($\sim$ 700 nm).

NEET Pattern MCQ Tips

• Biot-Savart: $B$ at centre of loop or straight wire — direct plug-in.
• Ampere's law: solenoid and toroid.
• Charged particle in B: $r = mv/(qB)$, $T$ independent of $v$.
• Force on parallel wires: attractive if same direction.
• Torque on loop: $\tau = NIAB\sin\theta$.
• Lenz's law: direction of induced current.
• Motional EMF: $\varepsilon = BLv$.
• Inductor stored energy: $\tfrac{1}{2}LI^2$.
• AC: RMS = peak/$\sqrt{2}$.
• LCR resonance: $\omega_0 = 1/\sqrt{LC}$, $Z = R$, $\cos\phi = 1$.
• Transformer: voltage ratio = turn ratio.
• EM spectrum: ordering by frequency/wavelength.

Common Confusions and Traps

• Magnetic force does no work — never increases KE of the particle.
• The period in cyclotron is independent of speed and radius — but the cyclotron breaks down at relativistic speeds.
• Lenz's law gives direction; Faraday gives magnitude.
• Inductor opposes change in current, not the current itself.
• In pure inductive or capacitive AC circuit, average power is zero (wattless).
• $\cos\phi$ may be increased toward 1 by adding capacitor in parallel (power-factor correction).
• Transformer cannot work with DC (no $d\Phi/dt$).
• EM waves carry momentum and exert radiation pressure.
• For a moving charge in $\vec B$, the path is a circle if $\vec v \perp \vec B$, helix if not.

Quick Revision Card

• $B$(infinite wire) $= \mu_0 I/(2\pi a)$.
• $B$(loop centre) $= \mu_0 I/(2R)$.
• Solenoid: $B = \mu_0 n I$ inside.
• Cyclotron $r = mv/(qB)$, $T = 2\pi m/(qB)$.
• Two parallel wires: $F/L = \mu_0 I_1 I_2/(2\pi d)$, attractive if same direction.
• Torque on loop: $\tau = NIAB\sin\theta$.
• $\varepsilon = -d\Phi/dt$; motional $\varepsilon = BLv$.
• Self-inductance solenoid: $L = \mu_0 n^2 A\ell$; energy $\tfrac{1}{2}LI^2$.
• RMS = peak$/\sqrt{2}$; average over half cycle = $2V_0/\pi$.
• $X_L = \omega L$, $X_C = 1/(\omega C)$.
• LCR: $Z = \sqrt{R^2 + (X_L - X_C)^2}$.
• Resonance: $\omega_0 = 1/\sqrt{LC}$, $\cos\phi = 1$.
• Transformer: $V_p/V_s = N_p/N_s$.
• $c = 1/\sqrt{\mu_0\varepsilon_0}$; $E_0/B_0 = c$.
• EM spectrum: $\gamma$, X, UV, vis, IR, microwave, radio (decreasing freq).

Worked NEET Examples

Example 1: Field at Centre of a Loop

Single loop of radius 5 cm carrying 2 A:

Example 2: Cyclotron Frequency

Proton in field $B = 1$ T:

Example 3: Force on Wire

A wire of length 20 cm carrying 5 A makes 30° angle with magnetic field 0.4 T:

Example 4: Motional EMF

A rod of length 0.5 m moves perpendicular to a field $B = 0.2$ T at 4 m/s:

Example 5: LCR Resonance

L = 10 mH, C = 1 μF. Resonance frequency:

Derivations

Magnetic Field on Axis of Circular Loop

A loop of radius $R$ in xy-plane, current $I$. By Biot-Savart, at axial distance $x$:

where $r = \sqrt{R^2 + x^2}$ and $\cos\alpha = R/r$ gives the axial component. Integrating around the loop ($\int dL = 2\pi R$):

At centre ($x = 0$): $B = \mu_0 I/(2R)$.

Force Between Two Parallel Wires

Wire 1 at origin with current $I_1$ creates field at wire 2 (distance $d$): $B_1 = \mu_0 I_1/(2\pi d)$.

Force per unit length on wire 2: $F/L = B_1 I_2 = \mu_0 I_1 I_2/(2\pi d)$.

Attractive when currents in same direction.

Self-Inductance of a Solenoid

A solenoid of length $\ell$, $N$ total turns, area $A$. Field inside: $B = \mu_0 (N/\ell) I = \mu_0 n I$. Flux through one turn: $\Phi_1 = BA$. Total flux linkage: $N\Phi_1 = N\mu_0 n I A = \mu_0 n^2 I A \ell$.

Self-inductance: $L = N\Phi_1/I = \mu_0 n^2 A\ell$.

EMF in a Rotating Coil

A coil of $N$ turns, area $A$, rotating with angular velocity $\omega$ in field $B$ has flux:

EMF:

Peak EMF: $\varepsilon_0 = NBA\omega$. This is the basis of an AC generator.

Impedance of LCR

Voltage phasor magnitude across R: $IR$. Across L: $I X_L = I\omega L$, leads I by 90°. Across C: $I X_C = I/(\omega C)$, lags I by 90°.

Net voltage: $V_0 = I_0 \sqrt{R^2 + (X_L - X_C)^2} = I_0 Z$, where $Z = \sqrt{R^2 + (X_L - X_C)^2}$ is impedance.

Phase angle: $\tan\phi = (X_L - X_C)/R$.

Resonance Condition

At $X_L = X_C$, $\omega L = 1/(\omega C)$, so $\omega_0 = 1/\sqrt{LC}$.

At this frequency, $Z = R$ (minimum), current is maximum, and $V$ is in phase with $I$ ($\cos\phi = 1$, max power).

Energy in Magnetic Field

Energy stored in inductor: $U = (1/2) L I^2$. Per unit volume:

For a solenoid with $B = \mu_0 n I$, volume $V = A\ell$:

Consistent with $L = \mu_0 n^2 A\ell$.

Eddy Currents and Their Applications

When a conductor moves through a changing field, induced currents (eddies) circulate within it. By Lenz's law, they oppose the motion (magnetic braking).

Applications:

• Electromagnetic brakes in trains.
• Damping in galvanometers.
• Induction heating.
• Metal detectors.

Drawbacks:

• Energy loss in transformer cores → laminated cores reduce this.

AC Power Details

Instantaneous power: $p(t) = v(t) i(t) = V_0 I_0 \sin(\omega t) \sin(\omega t - \phi)$.

Average over a cycle: $\bar P = V_\text{rms} I_\text{rms} \cos\phi$.

• $\phi = 0$ (purely resistive): max power.
• $\phi = \pi/2$ (purely L or C): zero average power.

Power factor $\cos\phi$ is improved by adding a capacitor in parallel with inductive load (common in industrial circuits to reduce reactive current).

Transformer Detailed Working

Two coils (primary and secondary) wound on a common iron core. Alternating current in primary creates alternating flux in core, which links secondary. By Faraday, EMF in secondary $\propto N_s d\Phi/dt$.

For ideal transformer (no losses):

• $V_s/V_p = N_s/N_p$ (voltage ratio = turn ratio).
• $I_s/I_p = N_p/N_s$ (current inverse).
• $P_s = P_p$ (power conservation).

Why iron core? High permeability concentrates flux; lamination reduces eddy currents.

EM Spectrum Production and Uses

Formula Sheet