Class 12 Physics Notes Chapter 6 (Electromagnetic induction) – Physics Part-I Book
Detailed Notes with MCQs of Chapter 6: Electromagnetic Induction. This is a crucial chapter, not just for your board exams but also extensively tested in various government recruitment exams involving physics. It bridges electricity and magnetism, explaining how changing magnetic fields can produce electric currents. Pay close attention to the concepts, laws, and formulas.
Chapter 6: Electromagnetic Induction - Detailed Notes
1. Introduction:
- Phenomenon discovered independently by Michael Faraday (UK) and Joseph Henry (USA) around 1830.
- Electromagnetic Induction (EMI): The phenomenon of producing an electromotive force (EMF), and hence an electric current (if the circuit is closed), in a conductor when the magnetic flux linked with it changes.
- The induced EMF and current last only as long as the change in magnetic flux continues.
2. Magnetic Flux (ΦB):
- It represents the number of magnetic field lines passing normally through a given area.
- For a uniform magnetic field B passing through a planar area A, the magnetic flux is given by:
ΦB = B ⋅ A = BA cos θ
where θ is the angle between the magnetic field vector B and the area vector A (normal to the surface). - Scalar quantity.
- SI Unit: Weber (Wb). 1 Wb = 1 Tesla-meter² (Tm²).
- CGS Unit: Maxwell (Mx). 1 Wb = 10⁸ Mx.
- Magnetic flux can be changed by changing:
- Magnitude of the magnetic field (B)
- Area of the loop (A)
- Angle (θ) between B and A.
3. Faraday's Laws of Electromagnetic Induction:
- First Law (Qualitative): Whenever the amount of magnetic flux linked with a circuit changes, an EMF is induced in the circuit.
- Second Law (Quantitative): The magnitude of the induced EMF (ε) in a circuit is directly proportional to the rate of change of magnetic flux linked with the circuit.
|ε| = |dΦB / dt| - For a coil with N closely wound turns, the total induced EMF is:
|ε| = N |dΦB / dt|
where ΦB is the flux linked with each turn.
4. Lenz's Law:
- This law gives the direction of the induced EMF and current.
- Statement: The polarity of the induced EMF (and hence the direction of the induced current) is such that it tends to oppose the change in magnetic flux that produced it.
- Mathematically incorporated into Faraday's second law with a negative sign:
ε = - dΦB / dt (for a single loop)
ε = - N dΦB / dt (for N turns) - Explanation:
- If flux is increasing, the induced current creates a magnetic field opposing the increase (i.e., opposite to the original field).
- If flux is decreasing, the induced current creates a magnetic field supporting the decrease (i.e., in the same direction as the original field).
- Lenz's Law and Conservation of Energy: Lenz's law is a direct consequence of the principle of conservation of energy. To induce a current, work must be done against the opposing force described by Lenz's law. This mechanical work gets converted into electrical energy (manifested as heat dissipation I²R if the circuit is closed).
5. Motional Electromotive Force (Motional EMF):
- EMF induced due to the motion of a conductor in a magnetic field.
- Consider a straight conductor of length 'l' moving with velocity 'v' perpendicular to a uniform magnetic field 'B', such that B, l, and v are mutually perpendicular.
- Origin: The Lorentz force (F = q(v x B)) acts on the free charge carriers (electrons) inside the conductor. This force drives electrons to one end, creating a potential difference.
- Magnitude of Induced EMF:
ε = Blv - Induced Current (if circuit closed with resistance R): I = ε / R = Blv / R
- Force on the conductor due to induced current: F = IlB = (Blv/R)lB = B²l²v / R. This force opposes the motion (as per Lenz's Law).
- Power required to maintain motion: P_mech = Fv = (B²l²v / R)v = B²l²v² / R
- Electrical power dissipated as heat: P_elec = I²R = (Blv/R)² R = B²l²v² / R
(P_mech = P_elec, confirming energy conservation). - General Case (arbitrary orientation): ε = ∫ (v x B) ⋅ dl
6. Eddy Currents (Foucault Currents):
- When a bulk piece of conductor (like a metal plate) is subjected to a changing magnetic flux, induced currents are set up within the body of the conductor in the form of closed loops resembling eddies or whirlpools.
- These currents oppose the change in flux (Lenz's Law).
- Disadvantages: They cause significant energy loss due to heating (I²R loss) in devices like transformer cores, armatures of motors/dynamos.
- Minimization: Using laminated cores (thin sheets insulated from each other) restricts the paths of eddy currents, increasing resistance and reducing their magnitude.
- Applications:
- Magnetic Braking: In trains, roller coasters (strong electromagnets induce eddy currents in rails/wheels, creating opposing force).
- Electromagnetic Damping: In galvanometers (eddy currents in the metallic frame damp oscillations).
- Induction Furnace: To produce high temperatures for melting metals (large eddy currents induced in the metal).
- Electric Power Meters: The rotating shiny disc utilizes eddy currents for measurement.
- Induction Motors.
- Speedometers.
7. Inductance:
-
The property of an electrical circuit by virtue of which it opposes any change in the current flowing through it. It's analogous to inertia in mechanics.
-
Inductance arises because a current itself produces a magnetic field and hence a magnetic flux linked with the circuit. A change in current causes a change in this self-flux, inducing a 'back EMF' that opposes the change.
-
a) Mutual Inductance (M):
- The phenomenon where a change of current in one coil induces an EMF in a neighbouring coil.
- Flux linked with coil 2 (Φ₂) due to current I₁ in coil 1: Φ₂ ∝ I₁ ⇒ Φ₂ = M₂₁ I₁
- Induced EMF in coil 2: ε₂ = - dΦ₂ / dt = - M₂₁ dI₁ / dt
- Similarly, flux linked with coil 1 (Φ₁) due to current I₂ in coil 2: Φ₁ = M₁₂ I₂
- Induced EMF in coil 1: ε₁ = - dΦ₁ / dt = - M₁₂ dI₂ / dt
- Reciprocity Theorem: M₁₂ = M₂₁ = M (Mutual inductance is the same for the pair).
- SI Unit: Henry (H). 1 H = 1 Wb/A = 1 Vs/A.
- Depends on: Size, shape, number of turns of the coils; their relative separation and orientation; permeability of the medium.
- Example: Mutual inductance of two long coaxial solenoids.
-
b) Self-Inductance (L):
- The property of a single coil by virtue of which it opposes a change of current flowing through itself.
- Flux linked with the coil (Φ) due to its own current (I): Φ ∝ I ⇒ Φ = LI
- Induced EMF (back EMF) in the coil: ε = - dΦ / dt = - L dI / dt
- SI Unit: Henry (H).
- Depends on: Geometry (size, shape, number of turns) of the coil; permeability of the core material.
- Example: Self-inductance of a long solenoid: L = μ₀ n² A l = μ₀ N² A / l
where n = N/l is the number of turns per unit length, N is the total number of turns, A is the cross-sectional area, l is the length, μ₀ is the permeability of free space. If a core with relative permeability μr is used, L = μ₀ μr n² A l.
-
8. Energy Stored in an Inductor:
- Work has to be done against the back EMF (ε = -L dI/dt) to establish a current in an inductor. This work is stored as magnetic potential energy (UB).
- Rate of work done: dW/dt = |ε|I = (L dI/dt) I
- Total work done to build current from 0 to I: W = ∫ dW = ∫₀ᴵ L I dI = ½ LI²
- Magnetic Potential Energy Stored: UB = ½ LI²
- Magnetic Energy Density (uB): Energy stored per unit volume. For a solenoid:
uB = UB / (Volume) = (½ LI²) / (Al) = ½ (μ₀n²Al) I² / (Al) = ½ μ₀ n² I²
Since B = μ₀nI inside a solenoid, I = B / (μ₀n)
uB = ½ μ₀ n² (B / μ₀n)² = B² / (2μ₀) (This is a general result for energy density in a magnetic field in vacuum/air).
9. AC Generator (Alternator/Dynamo):
- Principle: Electromagnetic Induction.
- Function: Converts mechanical energy into electrical energy (alternating current/EMF).
- Construction:
- Armature: Rectangular coil (ABCD) with many turns wound on a soft iron core.
- Field Magnets: Strong permanent magnets or electromagnets providing a uniform magnetic field.
- Slip Rings (R₁, R₂): Two metallic rings connected to the ends of the armature coil, rotating with it.
- Brushes (B₁, B₂): Carbon brushes kept pressed against the slip rings, providing external connection.
- Working: When the armature coil rotates in the magnetic field, the magnetic flux linked with it changes continuously. According to Faraday's law, an EMF is induced.
- Theory: Let the coil rotate with constant angular velocity ω. The angle θ between the magnetic field B and the area vector A at time t is θ = ωt (assuming θ=0 at t=0).
Flux at time t: ΦB = NBA cos θ = NBA cos(ωt)
Induced EMF: ε = - dΦB / dt = - d/dt (NBA cos(ωt)) = - NBA (-sin(ωt)) ω
ε = NBAω sin(ωt) - This is an alternating EMF. It can be written as:
ε = ε₀ sin(ωt)
where ε₀ = NBAω is the peak value or amplitude of the induced EMF. - The induced current is also alternating: I = ε / R = (ε₀ / R) sin(ωt) = I₀ sin(ωt), where I₀ = ε₀ / R is the peak current.
Multiple Choice Questions (MCQs):
-
The magnetic flux linked with a coil (in Wb) is given by the equation Φ = 5t² + 3t + 16. The magnitude of induced EMF in the coil at the fourth second will be:
(a) 10 V
(b) 43 V
(c) 108 V
(d) 16 V -
Lenz's law is a consequence of the law of conservation of:
(a) Charge
(b) Mass
(c) Momentum
(d) Energy -
A conducting rod of length 'l' is moving with velocity 'v' parallel to a long straight wire carrying current 'I'. The rod is perpendicular to the wire. The induced EMF in the rod is:
(a) B l v (where B is uniform)
(b) μ₀ I v / (2π)
(c) μ₀ I v l / (2πr) (where r is distance)
(d) Zero -
The SI unit of mutual inductance is:
(a) Weber
(b) Tesla
(c) Henry
(d) Farad -
Eddy currents are produced when:
(a) A metal is kept in varying magnetic field
(b) A metal is kept in steady magnetic field
(c) A circular coil is placed in a magnetic field
(d) Current is passed through a circular coil -
The self-inductance L of a solenoid of length l and area of cross-section A, with a fixed number of turns N, increases as:
(a) l and A increase
(b) l decreases and A increases
(c) l increases and A decreases
(d) both l and A decrease -
An AC generator generates an EMF given by ε = 314 sin(100πt) V. The frequency of the AC voltage is:
(a) 50 Hz
(b) 100 Hz
(c) 157 Hz
(d) 314 Hz -
A metallic ring is attached to the wall of a room. When the north pole of a magnet is brought near the ring, the direction of induced current in the ring will be:
(a) Clockwise
(b) Anticlockwise
(c) Towards North
(d) Towards South -
Energy stored in a pure inductor of inductance L carrying a steady current I is given by:
(a) LI²
(b) ½ LI²
(c) LI / 2
(d) L²I / 2 -
To reduce energy losses due to eddy currents in the core of a transformer:
(a) The core is made of a solid piece of soft iron
(b) The core is made of laminated sheets of soft iron
(c) The core is made of copper
(d) Air gaps are provided in the core
Answer Key for MCQs:
- (b) [ε = -dΦ/dt = -(10t + 3). At t=4s, |ε| = |-(10*4 + 3)| = 43 V]
- (d)
- (d) [The motion is parallel to the wire, so the velocity vector is parallel to the magnetic field lines (circles around the wire). Also, no change in flux through any loop formed with the rod.]
- (c)
- (a)
- (b) [L = μ₀ N² A / l. L increases if A increases and l decreases.]
- (a) [Comparing with ε = ε₀ sin(ωt), we have ω = 100π. Since ω = 2πf, f = ω / 2π = 100π / 2π = 50 Hz.]
- (b) [Bringing N-pole increases flux into the ring. By Lenz's law, induced current creates an opposing N-pole, which requires anticlockwise current when viewed from the magnet side.]
- (b)
- (b)
Make sure you understand the underlying principles behind each concept and formula. Practice numerical problems based on Faraday's Law, motional EMF, and inductance. Good luck with your preparation!