Electromagnetic Induction & AC Formula Sheet
All key Electromagnetic Induction and AC formulas - Faraday's law, motional EMF, inductance, LCR resonance, transformers. JEE Main & Advanced quick revision.
Every formula you need for Electromagnetic Induction and AC circuits, grouped by sub-topic for last-minute revision. Headline results are boxed.
Faraday’s Law & Magnetic Flux
The induced EMF equals the negative rate of change of flux linkage.
$$\boxed{\mathcal{E} = -N\frac{d\Phi_B}{dt}}$$Magnetic flux through a surface:
$$\Phi_B = \int \vec{B}\cdot d\vec{A} = BA\cos\theta$$| Quantity | Formula | Notes |
|---|---|---|
| Magnetic flux | $\Phi_B = BA\cos\theta$ | $\theta$ = angle between $\vec{B}$ and area vector $\vec{A}$ |
| Induced EMF | $\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$ | $N$ = number of turns |
| Flux linkage | $N\Phi_B$ | total flux linked with $N$ turns |
| Rotating coil flux | $\Phi_B = NBA\cos(\omega t)$ | coil spinning at angular velocity $\omega$ |
| Rotating coil EMF | $\mathcal{E} = NBA\omega\sin(\omega t)$ | AC generator principle |
| Max EMF (AC generator) | $\mathcal{E}_{max} = NBA\omega$ | peak value |
Lenz’s Law
The induced current always opposes the change in flux that produced it - this is the source of the negative sign in Faraday’s law.
| Statement | Meaning |
|---|---|
| Flux increasing ($\frac{d\Phi_B}{dt} > 0$) | $\mathcal{E} < 0$; induced field opposes external field |
| Flux decreasing ($\frac{d\Phi_B}{dt} < 0$) | $\mathcal{E} > 0$; induced field supports external field |
| Magnet approaching coil | near face becomes same pole → repulsion |
| Magnet receding from coil | near face becomes opposite pole → attraction |
The opposing force means you must do mechanical work to change the flux, and that work becomes the electrical energy: Mechanical work input = Electrical energy output. A magnet dropped through a coil falls slower than g because of this braking.
Motional EMF
For a straight conductor of length $\ell$ moving with velocity $v$ perpendicular to field $B$:
$$\boxed{\mathcal{E} = B\ell v}$$| Quantity | Formula | Notes |
|---|---|---|
| Straight rod (perpendicular) | $\mathcal{E} = B\ell v$ | “BLV” - $B$, $v$, $\ell$ mutually perpendicular |
| General angle | $\mathcal{E} = B\ell v\sin\theta$ | $\theta$ = angle between $\vec{v}$ and $\vec{B}$ |
| Vector form | $\mathcal{E} = \displaystyle\int (\vec{v}\times\vec{B})\cdot d\vec{\ell}$ | for non-uniform cases |
| Rotating rod (about one end) | $\mathcal{E} = \frac{1}{2}B\omega L^2$ | integrate, since $v = \omega r$ varies |
| Rotating disc (centre to rim) | $\mathcal{E} = \frac{1}{2}B\omega R^2$ | disc generator |
| Force on current-carrying rod | $F = BI\ell = \dfrac{B^2\ell^2 v}{R}$ | retarding force (Lenz) |
| Induced current (rod on rails) | $I = \dfrac{B\ell v}{R}$ | closed loop, resistance $R$ |
| Power dissipated | $P = \dfrac{(B\ell v)^2}{R} = F_{app}\,v$ | equals applied mechanical power |
Rod released from rest (velocity decay) and terminal velocity on an incline:
$$v(t) = v_0\,e^{-\frac{B^2\ell^2}{mR}t}, \qquad \tau = \frac{mR}{B^2\ell^2}, \qquad v_t = \frac{mgR\sin\theta}{B^2\ell^2}$$Self Inductance
$$\boxed{N\Phi_B = LI \qquad \mathcal{E} = -L\frac{dI}{dt}}$$| Quantity | Formula | Notes |
|---|---|---|
| Definition | $L = \dfrac{N\Phi_B}{I}$ | unit: Henry (H) = Wb/A = V·s/A |
| Back-EMF | $\mathcal{E} = -L\dfrac{dI}{dt}$ | proportional to rate of change of $I$ |
| Solenoid | $L = \dfrac{\mu_0 N^2 A}{\ell}$ | $L\propto N^2$; air core $\mu_0 = 4\pi\times10^{-7}$ H/m |
| Solenoid (material core) | $L = \dfrac{\mu_r\mu_0 N^2 A}{\ell}$ | $\mu_r$ = relative permeability |
| Toroid | $L = \dfrac{\mu_0 N^2 A}{2\pi r}$ | $r$ = mean radius |
| Energy stored | $U = \frac{1}{2}LI^2$ | energy in magnetic field |
| Energy density | $u = \dfrac{B^2}{2\mu_0}$ | per unit volume |
LR circuit (time constant $\tau = L/R$):
$$\text{Growth: } I(t) = I_0\left(1 - e^{-t/\tau}\right), \quad I_0 = \frac{\mathcal{E}_0}{R} \qquad\qquad \text{Decay: } I(t) = I_0\,e^{-t/\tau}$$Mutual Inductance
$$\boxed{\mathcal{E}_2 = -M\frac{dI_1}{dt} \qquad M = k\sqrt{L_1 L_2}}$$| Quantity | Formula | Notes |
|---|---|---|
| Definition | $M = \dfrac{N_2\Phi_{21}}{I_1} = \dfrac{N_1\Phi_{12}}{I_2}$ | reciprocity: $M_{12} = M_{21} = M$ |
| Induced EMF | $\mathcal{E}_2 = -M\dfrac{dI_1}{dt}$ | unit: Henry (H) |
| Coupling coefficient | $M = k\sqrt{L_1 L_2}$ | $0 \le k \le 1$; $k=1$ is perfect coupling |
| Solenoid pair | $M = \dfrac{\mu_0 N_1 N_2 A}{\ell}$ | $M\propto N_1 N_2$ |
| Energy of coupled coils | $U = \frac{1}{2}L_1 I_1^2 + \frac{1}{2}L_2 I_2^2 \pm M I_1 I_2$ | $+$ aiding, $-$ opposing |
Series combinations:
$$L_{eq} = L_1 + L_2 + 2M \ \text{(aiding)} \qquad L_{eq} = L_1 + L_2 - 2M \ \text{(opposing)} \qquad M = \frac{L_{aiding} - L_{opposing}}{4}$$Parallel combinations (general, and identical coils $L_1=L_2=L$):
$$\frac{1}{L_{eq}} = \frac{1}{L_1 \pm M} + \frac{1}{L_2 \pm M} \qquad\Rightarrow\qquad L_{eq} = \frac{L \pm M}{2}$$AC Circuits: R, L, C
AC voltage and current, with RMS (effective) values:
$$V = V_0\sin(\omega t), \qquad V_{rms} = \frac{V_0}{\sqrt{2}}, \qquad I_{rms} = \frac{I_0}{\sqrt{2}}, \qquad \omega = 2\pi f = \frac{2\pi}{T}$$| Element | Phase relation | Reactance / opposition | $I_0$ |
|---|---|---|---|
| Pure R | $V$ and $I$ in phase | $R$ (constant) | $I_0 = \dfrac{V_0}{R}$ |
| Pure L | $V$ leads $I$ by 90° | $X_L = \omega L = 2\pi f L$ ($\propto f$) | $I_0 = \dfrac{V_0}{X_L}$ |
| Pure C | $I$ leads $V$ by 90° | $X_C = \dfrac{1}{\omega C} = \dfrac{1}{2\pi f C}$ ($\propto \frac{1}{f}$) | $I_0 = \dfrac{V_0}{X_C}$ |
In an inductor (L): voltage E leads current I → ELI. In a capacitor (C): current I leads voltage E → ICE. Average power over a cycle for a pure inductor or capacitor is zero - they only store and return energy. Indian mains: $V_{rms} = 230$ V → $V_0 = 230\sqrt{2} \approx 325$ V at 50 Hz.
Series LCR Circuit & Resonance
$$\boxed{Z = \sqrt{R^2 + (X_L - X_C)^2} \qquad \tan\phi = \frac{X_L - X_C}{R}}$$| Quantity | Formula | Notes |
|---|---|---|
| Impedance | $Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$ | AC equivalent of resistance |
| Current | $I = \dfrac{V}{Z}$ | same current through all elements |
| Phase angle | $\tan\phi = \dfrac{X_L - X_C}{R}$ | $X_L>X_C$ inductive; $X_L |
| Average power | $P = V_{rms}I_{rms}\cos\phi = I_{rms}^2 R$ | only $R$ dissipates power |
| Power factor | $\cos\phi = \dfrac{R}{Z}$ | unity at resonance |
Resonance ($X_L = X_C$):
$$\omega_r = \frac{1}{\sqrt{LC}}, \qquad f_r = \frac{1}{2\pi\sqrt{LC}}$$At resonance: $Z = R$ (minimum), $I = \dfrac{V}{R}$ (maximum), $\phi = 0$, $\cos\phi = 1$, power maximum $P = \dfrac{V_{rms}^2}{R}$.
Quality factor and bandwidth:
$$Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r C R} = \frac{1}{R}\sqrt{\frac{L}{C}}, \qquad \Delta\omega = \frac{\omega_r}{Q}, \qquad V_L = V_C = QV \ \text{(at resonance)}$$| Property | Series LCR | Parallel LCR |
|---|---|---|
| Resonance condition | $X_L = X_C$ | $X_L = X_C$ |
| Resonant frequency | $f_r = \frac{1}{2\pi\sqrt{LC}}$ | $f_r = \frac{1}{2\pi\sqrt{LC}}$ |
| At resonance | $Z$ minimum, $I$ maximum | $Z$ maximum, $I$ minimum |
| Use | acceptor (passes $f_r$) | rejector (blocks $f_r$) |
Transformers
$$\boxed{\frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{I_1}{I_2} = K}$$| Quantity | Formula | Notes |
|---|---|---|
| Voltage ratio | $\dfrac{V_2}{V_1} = \dfrac{N_2}{N_1} = K$ | $K>1$ step-up, $K<1$ step-down |
| Current ratio | $\dfrac{I_2}{I_1} = \dfrac{N_1}{N_2} = \dfrac{1}{K}$ | voltage up → current down |
| Power conservation (ideal) | $V_1 I_1 = V_2 I_2$ | ideal transformer |
| Impedance transformation | $\dfrac{Z_1}{Z_2} = \dfrac{N_1^2}{N_2^2} = \dfrac{1}{K^2}$ | transforms as square of turns ratio |
| Efficiency | $\eta = \dfrac{P_{out}}{P_{in}} = \dfrac{V_2 I_2}{V_1 I_1}$ | typically 95-99% for power transformers |
| Copper loss | $P_{Cu} = I_1^2 R_1 + I_2^2 R_2$ | load-dependent ($\propto I^2$) |
| Iron loss | $P_{Fe} = P_{hysteresis} + P_{eddy}$ | hysteresis $\propto f$; eddy $\propto f^2 B^2$ |
| Max-efficiency condition | $P_{Cu} = P_{Fe}$ | variable losses = constant losses |
| Transmission loss | $P_{loss} = I^2 R$ | step up $V$ to cut $I$ and slash loss |
A transformer works only with AC. DC gives constant flux, so $\frac{d\Phi_B}{dt} = 0$ and no EMF is induced in the secondary. Laminated soft-iron cores reduce eddy-current losses.
Power-system voltage chain
graph LR
A["Power plant
11 kV"] -->|Step-up| B["Transmission
400 kV"]
B -->|Step-down| C["Substation
11 kV"]
C -->|Step-down| D["Home
230 V"]High transmission voltage means low current, so $P_{loss} = I^2 R$ drops drastically - the reason power is sent at 132-400 kV.
Master Formula Recap
| Topic | Headline formula |
|---|---|
| Faraday’s law | $\mathcal{E} = -N\frac{d\Phi_B}{dt}$ |
| Magnetic flux | $\Phi_B = BA\cos\theta$ |
| Motional EMF (rod) | $\mathcal{E} = B\ell v$ |
| Rotating rod | $\mathcal{E} = \frac{1}{2}B\omega L^2$ |
| Self inductance (solenoid) | $L = \frac{\mu_0 N^2 A}{\ell}$ |
| Inductor energy | $U = \frac{1}{2}LI^2$ |
| LR time constant | $\tau = \frac{L}{R}$ |
| Mutual inductance | $M = k\sqrt{L_1 L_2}$ |
| RMS value | $V_{rms} = \frac{V_0}{\sqrt{2}}$ |
| Reactances | $X_L = \omega L,\ X_C = \frac{1}{\omega C}$ |
| LCR impedance | $Z = \sqrt{R^2 + (X_L - X_C)^2}$ |
| Resonant frequency | $f_r = \frac{1}{2\pi\sqrt{LC}}$ |
| Quality factor | $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ |
| AC power | $P = V_{rms}I_{rms}\cos\phi$ |
| Transformer ratio | $\frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{I_1}{I_2}$ |
Revise the full chapter: Faraday’s Law · Lenz’s Law · Motional EMF · Self Inductance · Mutual Inductance · AC Circuits · LCR Circuits · Transformers