Comprehensive Study Notes on Electromagnetic Induction

Introduction and Historical Context

• Historical Separation: For a long period, electricity and magnetism were considered separate and unrelated phenomena.

• Early Developments: In the early 19th century (specifically the 1820s), experiments by Hans Christian Oersted and André-Marie Ampère established that moving electric charges (current) produce magnetic fields. This was demonstrated by the deflection of a magnetic compass needle near a current-carrying wire.

• The Inverse Question: Scientists questioned if the converse was possible: can moving magnets or changing magnetic fields produce electric currents?

• Discovery of Electromagnetic Induction (1830): Michael Faraday in England and Joseph Henry in the USA conducted experiments around 1830, conclusively proving that electric currents are induced in closed coils when subject to changing magnetic fields.

• Michael Faraday (1791–1867): Regarded as the greatest experimental scientist of the 19th century. His contributions include the discovery of electromagnetic induction, the laws of electrolysis, benzene, the invention of the electric motor, generator, and transformer, and the discovery of the rotation of the plane of polarization in an electric field.

• Joseph Henry (1797–1878): An American physicist at Princeton and the first director of the Smithsonian Institution. He improved electromagnets, invented an electromagnetic motor and a new telegraph, and discovered self-induction and mutual induction.

• Significance: This discovery led to the development of modern generators and transformers, forming the foundation of modern power systems.

Experiments of Faraday and Henry

Experiment 6.1: Bar Magnet and Coil

• Setup: A coil $C_1$ connected to a galvanometer $G$.

• Observations:

  • Pushing the North-pole of a bar magnet toward the coil causes a deflection in the galvanometer, indicating current.

  • The deflection lasts only as long as the magnet is in motion.

  • Pulling the magnet away causes a deflection in the opposite direction.

  • Using the South-pole results in deflections opposite to those observed with the North-pole for identical movements.

  • Higher speed of movement results in larger deflection/current.

  • Relative motion is the key; moving the coil toward a fixed magnet produces the same effects.

Experiment 6.2: Two-Coil Relative Motion

• Setup: Coil $C_1$ connected to a galvanometer, and a second coil $C_2$ connected to a battery.

• Mechanism: The steady current in $C_2$ produces a steady magnetic field.

• Observations:

  • Moving coil $C_2$ toward $C_1$ induces current in $C_1$.

  • Moving $C_2$ away reverses the direction of induced current.

  • Moving $C_1$ while $C_2$ is fixed produces identical induction results.

  • Induced current depends on the relative motion between the two coils.

Experiment 6.3: Stationary Coils and Changing Current

• Setup: Coil $C_1$ with galvanometer and Coil $C_2$ with a battery and a tapping key $K$; both coils are stationary.

• Observations:

  • Pressing key $K$: Momentary deflection in the galvanometer.

  • Holding key $K$: Current in $C_1$ drops to zero as the magnetic field from $C_2$ becomes constant.

  • Releasing key $K$: Momentary deflection in the opposite direction.

  • Enhancement: Deflection increases dramatically when an iron rod is inserted into the coils along their axis.

Magnetic Flux ($\Phi_B$)

• Definition: Magnetic flux through a surface of area $\mathbf{A}$ in a uniform magnetic field $\mathbf{B}$ is defined as the dot product:     $\Phi_B = \mathbf{B} \cdot \mathbf{A} = BA \cos(\theta)$

• Angle Variable: $\theta$ is the angle between the magnetic field vector $\mathbf{B}$ and the area vector $\mathbf{A}$.

• Non-uniform Fields: For curved surfaces or non-uniform fields, the surface is divided into small area elements $d\mathbf{A}_i$:     $\Phi_B = \sum_{all} \mathbf{B}_i \cdot d\mathbf{A}_i$

• Properties:

  • Quantity Type: Scalar.

  • SI Unit: Weber ($Wb$) or Tesla meter squared ($T\,m^2$).

  • Dimensions: $[M L^2 T^{-2} A^{-1}]$.

Faraday’s Law of Induction

• Principal Conclusion: An electromotive force (emf) is induced in a coil when the magnetic flux through the coil changes with time.

• General Law: The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.

• Mathematical Expression:     $\epsilon = -\frac{d\Phi_B}{dt}$

• Multiple Turns: For a closely wound coil of $N$ turns, where flux change is the same for each turn, the total emf is:     $\epsilon = -N \frac{d\Phi_B}{dt}$

• Increasing Induced EMF: Can be achieved by increasing the number of turns ($N$) or increasing the rate of change of flux ($\frac{d\Phi_B}{dt}$).

• Ways to Change Flux:

  1. Changing the magnetic field magnitude ($B$).

  2. Changing the shape or area of the coil ($A$) within the field.

  3. Changing the orientation ($\theta$) of the coil relative to the field.

Lenz’s Law and Conservation of Energy

• Lenz’s Law (1834): Formulated by Heinrich Friedrich Lenz. It states that the polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.

• Explanation via North-Pole Movement:

  • If a North-pole moves toward a coil, flux increases. The induced current creates a North-pole facing the magnet to repel its approach, thus opposing the flux increase.

  • If a North-pole leaves the coil, flux decreases. The induced current creates a South-pole facing the magnet to attract it, thus opposing the flux decrease.

• Conservation of Energy:

  • If the current did not oppose the motion (e.g., if it attracted an approaching North-pole), the magnet would accelerate indefinitely, creating energy from nothing, which violates the law of conservation of energy.

  • Because of the repulsive force, work must be done by an external agent to move the magnet. This mechanical work is dissipated as Joule heating produced by the induced current.

Motional Electromotive Force

• Scenario: A rod $PQ$ of length $l$ moves with velocity $v$ on rails $PQRS$ in a uniform magnetic field $B$ perpendicular to the plane.

• Flux Calculation: If $RQ = x$, flux $\Phi_B = Blx$.

• Derivation via Faraday's Law:     $\epsilon = -\frac{d\Phi_B}{dt} = -Bl \frac{dx}{dt}$     $\text{Since } \frac{dx}{dt} = -v, \text{ we get: } \epsilon = Blv$

• Lorentz Force Explanation:

  • A charge $q$ in moving rod $PQ$ experiences a force $F = qvB$ toward $Q$.

  • Work done moving the charge from $P$ to $Q$ is $W = Force \times distance = qvBl$.

  • Induced emf is the work per unit charge:         $\epsilon = \frac{W}{q} = Blv$

• Implication: A time-varying magnetic field generates an electric field. The electric fields produced by changing magnetic fields have different properties compared to those produced by static charges.

Inductance

• Definition: Inductance is the constant of proportionality between flux linkage ($N\Phi_B$) and current ($I$).

• Formula: $N\Phi_B = L I$ or $N\Phi_B = M I$.

• Physical Significance: Inductance is the electromagnetic analogue of mass (inertia) in mechanics. It opposes any change (growth or decay) of current.

• Quantity Type: Scalar.

• SI Unit: Henry ($H$).

• Dimensions: $[M L^2 T^{-2} A^{-2}]$.

Mutual Inductance ($M$)

• Concept: Flux in one coil caused by current in another coil.

• Co-axial Solenoids: For two long co-axial solenoids of length $l$, inner radius $r_1$, outer radius $r_2$, and turn densities $n_1, n_2$:     $M_{12} = M_{21} = M = \mu_0 n_1 n_2 \pi r_1^2 l$

• Medium Influence: With a material of relative permeability $\mu_r$:     $M = \mu_r \mu_0 n_1 n_2 \pi r_1^2 l$

• Induced EMF (Mutual):     $\epsilon_1 = -M \frac{dI_2}{dt}$

Self-Inductance ($L$)

• Concept: Flux linkage in a coil produced by its own current.

• Self-Induced EMF (Back EMF):     $\epsilon = -L \frac{dI}{dt}$

• Long Solenoid Calculation:     $L = \mu_0 n^2 Al$     $\text{With core: } L = \mu_r \mu_0 n^2 Al$

• Energy Stored in an Inductor:

  • Work must be done to establish current against back emf.

  • $W = \int L I \,dI = \frac{1}{2} L I^2$

• Magnetic Energy Density ($u_B$):     $u_B = \frac{B^2}{2 \mu_0}$

  • This is the magnetic analogue to electrostatic energy density $u_E = \frac{1}{2} \epsilon_0 E^2$.

AC Generator

• Developer: Nikola Tesla.

• Function: Converts mechanical energy into electrical energy using electromagnetic induction.

• Basic Components:

  • Armature (Coil): Mechanically rotated in a uniform magnetic field.

  • Field Magnets: Provide the magnetic flux.

  • Slip Rings and Brushes: Connect the rotating coil to the external circuit.

• Working Principle:

  • Angle between area vector and field changes as $\theta = \omega t$.

  • Flux $\Phi_B = BA \cos(\omega t)$.

  • Instantaneous emf for $N$ turns:         $\epsilon = NBA \omega \sin(\omega t)$

• Maximum EMF ($\epsilon_0$): Occurs when $\sin(\omega t) = 1$:     $\epsilon_0 = NBA \omega$     $\epsilon = \epsilon_0 \sin(2 \pi \nu t)$

  • Where $\nu$ is frequency (measured in $Hz$).

• Power Types:

  • Hydro-electric: Water power turns the armature.

  • Thermal: Steam from coal turns the armature.

  • Nuclear: Steam from nuclear reactions turns the armature.

• Commercial Standards: Frequency is $50\,Hz$ in India and $60\,Hz$ in the USA.

Numerical Examples and Problem Solutions

• Example 6.2 (Square Loop): Loop side $10\,cm$, resistance $0.5\,\Omega$, placed at $45^{\circ}$ to a field of $0.10\,T$ which drops to zero in $0.70\,s$.

  • Initial Flux $\Phi_{initial} = 0.1 \times 0.01 \times \cos(45^{\circ}) = 10^{-3} / \sqrt{2} \,Wb$.

  • Induced EMF $\epsilon = \frac{\Delta \Phi}{\Delta t} \approx 1.0\,mV$.

  • Induced Current $I = \frac{\epsilon}{R} = 2\,mA$.

• Example 6.3 (Circular Coil): Radius $10\,cm$, $500$ turns, resistance $2\,\Omega$, rotated $180^{\circ}$ in $0.25\,s$ in Earth's field ($3.0 \times 10^{-5}\,T$).

  • Initial Flux $\Phi_i = 3\pi \times 10^{-7}\,Wb$, Final Flux $\Phi_f = -3\pi \times 10^{-7}\,Wb$.

  • Average EMF $\epsilon = N \frac{\Delta \Phi}{\Delta t} = 3.8 \times 10^{-3}\,V$.

  • Average Current $I = 1.9 \times 10^{-3}\,A$.

• Example 6.6 (Rotating Rod): Length $1\,m$, frequency $50\,rev/s$, field $1\,T$.

  • $\epsilon = \frac{1}{2} B \omega R^2 = \frac{1}{2} \times 1.0 \times (2 \pi \times 50) \times 1^2 = 157\,V$.

• Example 6.7 (Wheel Spokes): $10$ spokes, length $0.5\,m$, $120\,rev/min$, field $0.4\,G$ ($0.4 \times 10^{-4}\,T$).

  • $\epsilon = \frac{1}{2} \omega B R^2 = 6.28 \times 10^{-5}\,V$.

  • Note: Number of spokes is irrelevant as they are in parallel.

• Example 6.10 (Bicycle Generator): $N=100$, $A=0.10\,m^2$, $\nu = 0.5\,Hz$, $B=0.01\,T$.

  • $\epsilon_0 = 100 \times 0.01 \times 0.1 \times 2 \pi \times 0.5 = 0.314\,V$.