Electromagnetic Induction and Alternating Currents — Revision Notes

⚡ 30-Second Revision

Magnetic Flux: — $\Phi_B = BA \cos\theta$ (Wb)\n- Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ (V)\n- Lenz's Law: Induced current opposes change in flux.\n- Motional EMF: $\mathcal{E} = Blv$ (V) (for perpendicular motion)\n- Self-Inductance: $\mathcal{E} = -L \frac{dI}{dt}$ (L in H)\n- Mutual Inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ (M in H)\n- Inductive Reactance: $X_L = \omega L = 2\pi f L$ ($\Omega$)\n- Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ ($\Omega$)\n- Impedance (Series RLC): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ ($\Omega$)\n- Resonant Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$ (rad/s) or $f_0 = \frac{1}{2\pi\sqrt{LC}}$ (Hz)\n- Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$\n- Average Power: $P_{avg} = V_{rms} I_{rms} \cos\phi$ (W)\n- Power Factor: $\cos\phi$\n- RMS Values: $V_{rms} = V_0/\sqrt{2}$, $I_{rms} = I_0/\sqrt{2}$\n- Ideal Transformer: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$

2-Minute Revision

Electromagnetic Induction (EMI) is the generation of EMF and current by changing magnetic flux. Faraday's Laws quantify this: induced EMF is proportional to the rate of change of magnetic flux ($\mathcal{E} = -N \frac{d\Phi_B}{dt}$).

Lenz's Law dictates the direction, stating the induced current opposes the change in flux, a consequence of energy conservation. Motional EMF ($Blv$) arises from a conductor moving in a magnetic field.

Self-induction ($L$) is a coil's opposition to current change in itself, while mutual induction ($M$) is when a changing current in one coil induces EMF in another. \nAlternating Current (AC) is generated by rotating a coil in a magnetic field, producing sinusoidal voltage and current.

In AC circuits, resistors (R) are in phase, inductors (L) have voltage leading current by $90^\circ$ ($X_L = \omega L$), and capacitors (C) have current leading voltage by $90^\circ$ ($X_C = 1/\omega C$).

The total opposition is impedance ($Z = \sqrt{R^2 + (X_L - X_C)^2}$). Resonance occurs when $X_L = X_C$, leading to maximum current at $\omega_0 = 1/\sqrt{LC}$. Average power is $P_{avg} = V_{rms} I_{rms} \cos\phi$, where $\cos\phi$ is the power factor.

Transformers use mutual induction to step up/down AC voltages efficiently for transmission.

5-Minute Revision

Start with the core concept of magnetic flux ($\Phi_B = BA \cos\theta$), the 'amount' of magnetic field passing through an area. The key to EMI is a *change* in this flux. Faraday's Laws state that an EMF is induced if flux changes, and its magnitude is proportional to the rate of change ($\mathcal{E} = -N \frac{d\Phi_B}{dt}$).

The crucial Lenz's Law provides the direction: the induced current creates a magnetic field that *opposes* the change in the original flux, ensuring energy conservation. For example, if a magnet approaches a coil, the induced current creates a like pole to repel it.

\nMotional EMF ($\mathcal{E} = Blv$) is a direct application, where a conductor moving in a field experiences a force on its charges, creating a potential difference. Self-induction ($\mathcal{E} = -L \frac{dI}{dt}$) is when a changing current in a coil induces an EMF in itself, opposing the current change.

Mutual induction ($\mathcal{E}_2 = -M \frac{dI_1}{dt}$) is similar, but involves two coils. \nAlternating Current (AC) is generated by rotating a coil in a magnetic field, causing the induced EMF to vary sinusoidally ($\mathcal{E} = NBA\omega \sin(\omega t)$).

This leads to AC, which periodically reverses direction. \nIn AC circuits, components behave differently: resistors (R) have voltage and current in phase. Inductors (L) introduce inductive reactance ($X_L = \omega L$), where voltage leads current by $90^\circ$.

Capacitors (C) introduce capacitive reactance ($X_C = 1/\omega C$), where current leads voltage by $90^\circ$. For a series RLC circuit, the total opposition is impedance ($Z = \sqrt{R^2 + (X_L - X_C)^2}$), and the phase difference is $\tan\phi = (X_L - X_C)/R$.

Resonance occurs when $X_L = X_C$, leading to minimum impedance ($Z=R$) and maximum current at the resonant frequency $\omega_0 = 1/\sqrt{LC}$. Average power in AC circuits is $P_{avg} = V_{rms} I_{rms} \cos\phi$, where $\cos\phi$ is the power factor.

Only the resistive part dissipates average power. Transformers are vital devices based on mutual induction, efficiently stepping up or stepping down AC voltages for power transmission using the relation $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$.

Remember RMS values ($V_{rms} = V_0/\sqrt{2}$) for power calculations. Practice problems on all these concepts, especially phase relationships and resonance.

Prelims Revision Notes

1
Magnetic Flux ($\Phi_B$): — Scalar, $\Phi_B = BA \cos\theta$. Unit: Weber (Wb). Change in flux is key for EMI.\n2. Faraday's Laws: Induced EMF $\mathcal{E} = -N \frac{d\Phi_B}{dt}$. Magnitude is proportional to rate of change of flux. $N$ is number of turns.\n3. Lenz's Law: Direction of induced current opposes the *change* in magnetic flux. Crucial for conceptual questions. Ensures energy conservation.\n4. Motional EMF: $\mathcal{E} = Blv$ for a conductor of length $l$ moving with velocity $v$ perpendicular to magnetic field $B$. If not perpendicular, use perpendicular components.\n5. Eddy Currents: Induced circulating currents in bulk conductors due to changing flux. Cause heating (energy loss). Reduced by laminating cores.\n6. Self-Inductance (L): Property of a coil to oppose change in current through itself. $\mathcal{E} = -L \frac{dI}{dt}$. Unit: Henry (H). $L = \frac{\mu_0 N^2 A}{l}$ for a solenoid.\n7. Mutual Inductance (M): Property where changing current in one coil induces EMF in a nearby coil. $\mathcal{E}_2 = -M \frac{dI_1}{dt}$. Unit: Henry (H).\n8. AC Generator: Converts mechanical to electrical energy. Principle: EMI. Produces sinusoidal EMF $\mathcal{E} = NBA\omega \sin(\omega t)$.\n9. AC Circuit Components:\n * Resistor (R): $V_R = I_R R$. Voltage and current are in phase.\n * Inductor (L): Inductive Reactance $X_L = \omega L = 2\pi f L$. Voltage leads current by $90^\circ$ (ELI).\n * Capacitor (C): Capacitive Reactance $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$. Current leads voltage by $90^\circ$ (ICE).\n10. Series RLC Circuit:\n * Impedance (Z): Total opposition to current. $Z = \sqrt{R^2 + (X_L - X_C)^2}$.\n * **Phase Angle ($\phi$):** $\tan\phi = \frac{X_L - X_C}{R}$.\n * Current: $I_{rms} = V_{rms}/Z$.\n11. Resonance: Occurs when $X_L = X_C$. Impedance is minimum ($Z=R$), current is maximum. Resonant angular frequency $\omega_0 = \frac{1}{\sqrt{LC}}$. Resonant frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$.\n12. Quality Factor (Q-factor): $Q = \frac{\omega_0 L}{R} = \frac{1}{R} \sqrt{\frac{L}{C}}$. Measures sharpness of resonance.\n13. Power in AC Circuits:\n * Instantaneous Power: $P = VI$.\n * Average Power: $P_{avg} = V_{rms} I_{rms} \cos\phi$. Only resistor dissipates average power.\n * Power Factor: $\cos\phi$. For pure R, $\cos\phi = 1$. For pure L or C, $\cos\phi = 0$.\n * RMS Values: $V_{rms} = V_0/\sqrt{2}$, $I_{rms} = I_0/\sqrt{2}$. These are effective values.\n14. Transformers: Devices to change AC voltage. Principle: Mutual Induction. For ideal transformer: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$. Power conserved: $V_p I_p = V_s I_s$. Only works with AC.

Vyyuha Quick Recall

For AC circuit phase relationships: ELI the ICE man\n- ELI: EMF (Voltage) Leads In current in an Lnductor (by $90^\circ$).\n- ICE: In current Capacitor EMF (Voltage) (Current Leads EMF in a Capacitor by $90^\circ$).\nThis helps remember which quantity leads or lags in L and C circuits.