⚡ Electromagnetic Induction & Alternating Current
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PART A electromagnetic induction
1Magnetic Flux
$$\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$$
\(\Phi_B\) = Magnetic flux (Wb) |
\(\vec{B}\) = Magnetic field (T) |
\(\vec{A}\) = Area vector (m²) |
\(\theta\) = Angle between \(\vec{B}\) and normal to area
TRICK Flux is maximum when \(\theta = 0°\) (\(\Phi = BA\)), minimum (zero) when \(\theta = 90°\).
2Faraday's Laws of Electromagnetic Induction
$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$
\(\mathcal{E}\) = Induced EMF (V) |
\(\Phi_B\) = Magnetic flux |
\(t\) = Time
For N turns:
$$\mathcal{E} = -N\frac{d\Phi_B}{dt}$$
\(N\) = Number of turns in the coil
TRICK \(\mathcal{E} = -N \frac{\Delta \Phi}{\Delta t}\) — Induced EMF depends on rate of change of flux, not flux itself.
3Induced EMF and Current
$$\mathcal{E} = \frac{\Delta \Phi}{\Delta t} \qquad I = \frac{\mathcal{E}}{R}$$
\(I\) = Induced current (A) |
\(R\) = Resistance of the circuit (Ω)
4Lenz's Law
The direction of induced EMF is such that it opposes the change in magnetic flux that produced it.
$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$
\((-)\) sign indicates opposition (Lenz's Law)
TRICK Lenz's Law = Conservation of Energy. The induced current always opposes the cause of induction.
5Eddy Currents
No formula — induced circulating currents in bulk conductors. Used in: Induction furnace, braking in trains, speedometer.
TRICK Eddy currents are minimised by laminating the core (thin insulated layers).
6Self-Inductance (L)
$$\mathcal{E} = -L\frac{dI}{dt} \qquad \Phi = LI$$
\(L\) = Self-inductance (H) |
\(I\) = Current |
\(\Phi\) = Flux linked with the coil
7Self-Inductance of a Solenoid
$$L = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}$$
\(n\) = Turns per unit length |
\(N\) = Total turns |
\(A\) = Area of cross-section |
\(l\) = Length of solenoid
8Mutual Inductance (M)
$$\mathcal{E}_2 = -M\frac{dI_1}{dt} \qquad \Phi_2 = MI_1$$
\(M\) = Mutual inductance (H) |
\(I_1\) = Current in primary coil |
\(\Phi_2\) = Flux linked with secondary coil
9Mutual Inductance of Two Coaxial Solenoids
$$M = \mu_0 n_1 n_2 A l$$
$$M = \frac{\mu_0 N_1 N_2 A}{l}$$
\(n_1, n_2\) = Turns per unit length of solenoids |
\(N_1, N_2\) = Total turns |
\(A\) = Area of cross-section |
\(l\) = Length
PART B alternating current
10Alternating Current (AC) — Peak and RMS Values
$$I = I_0 \sin\omega t \qquad V = V_0 \sin\omega t$$
\(I_0, V_0\) = Peak current/voltage |
\(\omega\) = Angular frequency (\(2\pi f\)) |
\(t\) = Time
$$I_{\text{RMS}} = \frac{I_0}{\sqrt{2}} \qquad V_{\text{RMS}} = \frac{V_0}{\sqrt{2}}$$
\(I_{\text{RMS}}, V_{\text{RMS}}\) = Root Mean Square values
TRICK RMS = Peak / √2. For domestic supply, \(220V\) is RMS value → Peak \(V_0 = 220\sqrt{2} \approx 311V\).
11Reactance and Impedance
$$X_L = \omega L = 2\pi f L \qquad X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
\(X_L\) = Inductive reactance (Ω) |
\(X_C\) = Capacitive reactance (Ω) |
\(f\) = Frequency (Hz)
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
\(Z\) = Impedance (Ω) |
\(R\) = Resistance (Ω)
12Phasor Diagrams
Pure Resistive Circuit: \(V\) and \(I\) in phase (\(\phi = 0°\))
Pure Inductive Circuit: \(I\) lags \(V\) by \(90°\) (\(\phi = +90°\))
Pure Capacitive Circuit: \(I\) leads \(V\) by \(90°\) (\(\phi = -90°\))
TRICK "ELI the ICE man" — E (Voltage) leads I in L (inductor); I leads E in C (capacitor).
13LR Circuit
$$Z = \sqrt{R^2 + X_L^2} \qquad \tan\phi = \frac{X_L}{R}$$
\(\phi\) = Phase angle between \(V\) and \(I\)
14CR Circuit
$$Z = \sqrt{R^2 + X_C^2} \qquad \tan\phi = -\frac{X_C}{R}$$
15LCR Series Circuit
$$Z = \sqrt{R^2 + (X_L - X_C)^2} \qquad \tan\phi = \frac{X_L - X_C}{R}$$
$$V = \sqrt{V_R^2 + (V_L - V_C)^2} \qquad V_R = IR, \quad V_L = IX_L, \quad V_C = IX_C$$
\(V_R, V_L, V_C\) = Voltage across R, L, C respectively
16Resonance in LCR Circuit
$$X_L = X_C \qquad \omega_r = \frac{1}{\sqrt{LC}} \qquad f_r = \frac{1}{2\pi\sqrt{LC}}$$
\(\omega_r\) = Resonant angular frequency |
\(f_r\) = Resonant frequency (Hz)
$$Z_{\text{min}} = R \qquad I_{\text{max}} = \frac{V}{R}$$
TRICK At resonance: \(Z = R\) (minimum), \(I = V/R\) (maximum), and circuit is purely resistive (\(\phi = 0°\)).
17LC Oscillator (Qualitative)
Energy oscillates between capacitor (electric field) and inductor (magnetic field). Frequency:
$$f = \frac{1}{2\pi\sqrt{LC}}$$
TRICK LC oscillation is the basis of radio transmitters and receivers.
18Power in AC Circuit
$$P = VI\cos\phi \qquad P_{\text{avg}} = V_{\text{RMS}} I_{\text{RMS}} \cos\phi$$
\(P\) = Average power (W) |
\(\cos\phi\) = Power factor
TRICK Power factor = \(\cos\phi = \frac{R}{Z}\). For pure resistor: 1 (max power). For pure L or C: 0 (zero power).
19Wattless Current
Current that does not consume power: \(I_{\text{wattless}} = I_{\text{RMS}} \sin\phi\)
TRICK In pure inductor/capacitor, \(\phi = 90°\), \(\cos\phi = 0\), so power = 0 — all current is wattless.
20AC Generator (Dynamo)
$$\mathcal{E} = \mathcal{E}_0 \sin\omega t \qquad \mathcal{E}_0 = NBA\omega$$
\(\mathcal{E}_0\) = Peak EMF |
\(N\) = Number of turns |
\(B\) = Magnetic field |
\(A\) = Area |
\(\omega\) = Angular speed of rotation
TRICK Generator = Mechanical energy → Electrical energy (based on Faraday's Law).
21Transformer
$$\frac{V_s}{V_p} = \frac{N_s}{N_p} \qquad \frac{I_s}{I_p} = \frac{N_p}{N_s}$$
\(V_p, V_s\) = Primary & Secondary voltages |
\(N_p, N_s\) = Primary & Secondary turns |
\(I_p, I_s\) = Primary & Secondary currents
$$P_p = P_s \quad (\text{ideal}) \qquad \text{Efficiency} = \frac{P_s}{P_p} \times 100\%$$
TRICK Step-up: \(N_s > N_p\), \(V_s > V_p\), \(I_s < I_p\). Step-down: \(N_s < N_p\), \(V_s < V_p\), \(I_s > I_p\).
📐 Units & Dimensions
| Quantity | SI Unit | Alternative / Equivalent | Dimension |
|---|---|---|---|
| Magnetic Flux \(\Phi\) | Wb (weber) | T·m², V·s | \(M\,L^2\,T^{-2}\,A^{-1}\) |
| EMF \(\mathcal{E}\) | V (volt) | Wb/s | \(M\,L^2\,T^{-3}\,A^{-1}\) |
| Self-Inductance \(L\) | H (henry) | V·s/A, Ω·s | \(M\,L^2\,T^{-2}\,A^{-2}\) |
| Mutual Inductance \(M\) | H (henry) | V·s/A | \(M\,L^2\,T^{-2}\,A^{-2}\) |
| Reactance \(X_L, X_C\) | Ω (ohm) | — | \(M\,L^2\,T^{-3}\,A^{-2}\) |
| Impedance \(Z\) | Ω (ohm) | — | \(M\,L^2\,T^{-3}\,A^{-2}\) |
| Power \(P\) | W (watt) | V·A | \(M\,L^2\,T^{-3}\) |
| Frequency \(f\) | Hz (s⁻¹) | — | \(T^{-1}\) |
| Angular frequency \(\omega\) | rad/s | s⁻¹ | \(T^{-1}\) |
| Current \(I\) | A (ampere) | C/s | \(A\) |
| Voltage \(V\) | V (volt) | J/C | \(M\,L^2\,T^{-3}\,A^{-1}\) |
⚡ Quick Recall
| Magnetic Flux | \(\Phi = BA\cos\theta\) |
| Faraday's Law | \(\mathcal{E} = -N\dfrac{d\Phi}{dt}\) |
| Self-Inductance | \(\mathcal{E} = -L\dfrac{dI}{dt}\) |
| Self-Inductance (Solenoid) | \(L = \dfrac{\mu_0 N^2 A}{l}\) |
| Mutual Inductance | \(\mathcal{E}_2 = -M\dfrac{dI_1}{dt}\) |
| Mutual Inductance (Coaxial) | \(M = \dfrac{\mu_0 N_1 N_2 A}{l}\) |
| RMS Value | \(I_{\text{RMS}} = \dfrac{I_0}{\sqrt{2}}\) |
| Inductive Reactance | \(X_L = \omega L\) |
| Capacitive Reactance | \(X_C = \dfrac{1}{\omega C}\) |
| Impedance (LCR) | \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) |
| Resonance Frequency | \(f_r = \dfrac{1}{2\pi\sqrt{LC}}\) |
| Power in AC | \(P = VI\cos\phi\) |
| Power Factor | \(\cos\phi = \dfrac{R}{Z}\) |
| Transformer | \(\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}\) |
| AC Generator | \(\mathcal{E}_0 = NBA\omega\) |
| LC Oscillator | \(f = \dfrac{1}{2\pi\sqrt{LC}}\) |
📘 EMI & AC · complete formula sheet with symbol meanings · updated 10 Aug 2026
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