"Electromagnetic Induction and Alternating Current"

About

If electricity can produce magnetism, can magnetism produce electricity? Yes — through electromagnetic induction. This chapter covers Faraday's laws, Lenz's law, self and mutual induction, and the principles of alternating current (AC). This phenomenon powers generators, transformers, and much of modern electrical technology.

Key Concepts

19.1 Faraday's Laws of Electromagnetic Induction

First Law: Whenever the magnetic flux linked with a circuit changes, an EMF is induced in it.

Second Law: The magnitude of induced EMF equals the rate of change of magnetic flux:

$|e|=N\frac{d{\Phi }_B}{dt}$

Where ${\Phi }_B=BA\cos \theta$ (magnetic flux).

19.2 Lenz's Law

The direction of induced current is such that it opposes the change in magnetic flux that produced it.

Lenz's law is a consequence of the conservation of energy. The negative sign in Faraday's law comes from Lenz's law:

$e=-N\frac{d{\Phi }_B}{dt}$

19.3 Induced EMF

EMF from changing magnetic field: If ${\Phi }_B=A+D{t}^2$, then $e=N(2D)t$

EMF depends on:

1. Number of turns ($N$)
2. Rate of change of flux ($d\Phi /dt$)
3. Orientation of coil ($\cos \theta$)

Maximum EMF: When the plane of the coil is perpendicular to $B$ ($\theta =0°$, $\cos \theta =1$)

Minimum EMF (zero): When the plane is parallel to $B$ ($\theta =90°$, $\cos \theta =0$)

19.4 Motional EMF

When a conductor of length $l$ moves with velocity $v$ perpendicular to field $B$:

$e=Blv$

19.5 Self and Mutual Inductance

Self-inductance ($L$): $e=-L\frac{dI}{dt}$

SI unit: henry (H). $L=\frac{N{\Phi }_B}{I}$

Mutual inductance ($M$): $e_2=-M\frac{d{I}_1}{dt}$

19.6 Alternating Current (AC)

AC voltage: $V=V_0\sin \omega t$

AC current: $I=I_0\sin (\omega t-\varphi )$

RMS values: $V_{\text{rms}}=\frac{V_0}{\sqrt{2}},I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$

Reactance:

• Inductive: $X_L=\omega L$
• Capacitive: $X_C=\frac{1}{\omega C}$

Impedance (RLC series): $Z=\sqrt{R^2+(X_L-X_C{)}^2}$

INTEXT QUESTIONS 19.1

Q1. A 1000 turn coil has a radius of 5 cm. Calculate the EMF developed across the coil if the magnetic field through the coil is reduced from 10 T to 0 in (a) 1 s, (b) 1 ms.

Ans: $A=\pi r^2=\pi (0.05{)}^2=0.0025\pi$ m² $\Delta \Phi =(0-10)\times 0.0025\pi =-0.025\pi$

(a) t = 1 s: $|e|=1000\times 0.025\pi /1\approx 78.5$ V

(b) t = 1 ms: $|e|=1000\times 0.025\pi /0.001\approx 78,500$ V

Q2. The magnetic flux linking each loop of a 250-turn coil is given by ${\varphi }_B(t)=A+D{t}^2$, where A = 3 Wb and D = 15 Wb/s². Show that (a) $e=(2ND)t$, and (b) evaluate the EMF at t = 0 s and t = 3.0 s.

Ans: (a) $e=N\frac{d\varphi }{dt}=N\cdot 2Dt=(2ND)t$

(b) At $t=0$: $e=0$ V. At $t=3$ s: $e=2\times 250\times 15\times 3=22,500$ V.

Q3. For a conducting loop of area S at angle $\theta$ to B, show $e=(dB/dt)S\cos \theta$. For what orientation is $e$ (a) maximum, (b) minimum?

Ans: $\Phi =BS\cos \theta$, so $e=(dB/dt)S\cos \theta$.

(a) Maximum: $\theta =0°$ (loop perpendicular to field, $\cos \theta =1$).

(b) Minimum (zero): $\theta =90°$ (loop parallel to field, $\cos \theta =0$).

INTEXT QUESTIONS 19.2

Q1. A bar magnet moves to the right. What is the sense of induced current in stationary loop A (magnet approaching) and loop B (magnet receding)?

Ans: Loop A: Magnet approaches → flux increases → induced current creates opposing field → anticlockwise (as viewed from approaching side). Loop B: Magnet recedes → flux decreases → induced current supports field → clockwise.

Q2. Inside an ideal solenoid, B is increasing. B = 0 outside. In which conducting loops is there induced current? What is the sense?

Ans: Loop inside solenoid: Flux is increasing → induced current flows, opposing the increase (direction depends on B direction). Loop outside solenoid: B = 0 and remains zero → no flux change → no induced current.

Terminal Exercise

1. State and explain Faraday's laws of electromagnetic induction.

2. State Lenz's law. Show that it follows from the law of conservation of energy.

3. A coil of 500 turns and area 0.01 m² is placed perpendicular to a magnetic field of 0.2 T. The field is reduced to zero in 0.1 s. Find the induced EMF.

4. Derive the expression for motional EMF: $e=Blv$.

5. Define self-inductance and mutual inductance. Derive the expression for self-inductance of a long solenoid.

6. Explain the principle and working of an AC generator with a labelled diagram.

7. A 50 Ω resistor, a 0.1 H inductor, and a 100 μF capacitor are connected in series to a 220 V, 50 Hz supply. Find: (a) impedance, (b) current, (c) power factor.

8. Define RMS value of AC. Show that $V_{\text{rms}}=V_0/\sqrt{2}$.

9. Explain the terms: (a) inductive reactance, (b) capacitive reactance. How do they vary with frequency?

10. A transformer has 1000 primary turns and 100 secondary turns. If the primary is connected to 220 V AC, find the secondary voltage. What type of transformer is this?

11. Why can't a transformer step up or step down DC voltage?

12. Describe an experiment to demonstrate electromagnetic induction using a coil and a bar magnet.

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