Electromagnetic Induction and Alternating Current
1. Introduction to Electromagnetic Induction
Electromagnetic Induction (EMI) is the phenomenon in which an electromotive force (emf)
is induced in a conductor when the magnetic flux linked with it changes.
2. Magnetic Flux
Magnetic flux through a surface is defined as the total number of magnetic field lines
passing through that surface.
$$\Phi = \vec{B} \cdot \vec{A} = BA\cos\theta$$
Unit of magnetic flux is Weber (Wb).
3. Faraday’s Laws of Electromagnetic Induction
First Law
Whenever the magnetic flux linked with a circuit changes, an emf is induced in the circuit.
Second Law
$$\mathcal{E} = -\frac{d\Phi}{dt}$$
Negative sign indicates direction of induced emf (explained by Lenz’s law).
4. Lenz’s Law
The direction of induced current is such that it opposes the cause that produces it.
Lenz’s law is a direct consequence of the law of conservation of energy.
5. Motional emf
When a conductor moves in a magnetic field, an emf is induced due to magnetic Lorentz force.
$$\mathcal{E} = B\ell v$$
6. Eddy Currents
Eddy currents are circulating currents induced in bulk conductors due to changing magnetic flux.
Applications:
- Magnetic braking
- Induction furnace
- Speedometers
7. Self Inductance
Self inductance is the property of a coil by which it opposes change in current flowing through it.
$$\mathcal{E} = -L\frac{dI}{dt}$$
8. Mutual Inductance
Mutual inductance is the phenomenon in which change of current in one coil induces emf in another coil.
$$\mathcal{E} = -M\frac{dI}{dt}$$
9. Energy Stored in an Inductor
$$U = \frac{1}{2}LI^2$$
10. Alternating Current (AC)
Alternating current is a current whose magnitude and direction change periodically with time.
$$i = I_0 \sin(\omega t)$$
11. RMS Value of AC
$$I_{\text{rms}} = \frac{I_0}{\sqrt{2}}, \quad
V_{\text{rms}} = \frac{V_0}{\sqrt{2}}$$
12. AC Through Resistor
$$V = IR$$
Current and voltage are in the same phase.
13. AC Through Inductor
$$X_L = \omega L$$
Current lags voltage by $90^\circ$.
14. AC Through Capacitor
$$X_C = \frac{1}{\omega C}$$
Current leads voltage by $90^\circ$.
15. LCR Series Circuit
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
16. Resonance in LCR Circuit
$$\omega_0 = \frac{1}{\sqrt{LC}}$$
At resonance: $X_L = X_C$, impedance is minimum.
17. Power in AC Circuit
$$P = V_{\text{rms}} I_{\text{rms}} \cos\phi$$
$\cos\phi$ is called power factor.
18. Transformer
A transformer is a device used to increase or decrease AC voltage using mutual induction.
$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
19. Step-Up and Step-Down Transformer
- Step-up: $V_s > V_p$
- Step-down: $V_s < V_p$
20. Common JEE Traps
- Forgetting negative sign in Faraday’s law
- Confusing RMS with peak values
- Wrong phase relationship
- Ignoring energy loss in transformers
21. Final Revision Checklist
You have mastered this chapter if you can:
- Apply Faraday and Lenz laws correctly
- Solve AC circuit problems
- Handle phasor diagrams mentally
- Understand resonance and transformers