Comprehensive Guide to Electromagnetic Induction

Fundamental Principles of Electromagnetic Induction

• Electromagnetic Induction Definition: This process occurs whenever the magnetic flux linked with a coil changes, resulting in an electromotive force (emf) being induced in the coil.

• Faraday’s Law of Electromagnetic Induction: This law states that the magnitude of the induced emf ($\epsilon$) is equal to the rate of change of magnetic flux ($\phi$). Mathematically, it is expressed as:     $\epsilon = \frac{d\phi}{dt}$

• Lenz’s Law: This law dictates the direction of the induced emf. It states that the direction of the induced emf is always such as to oppose the rate of change of magnetic flux that produced it. Lenz's law is represented by the formula:     $\epsilon = -\frac{d\phi}{dt}$     The negative sign explicitly indicates that the induced emf opposes the change in flux.

• Lenz’s Law and Energy Conservation: Lenz’s law is a statement of the conservation of energy. When a magnet is brought near a coil, an induced emf is developed, and the coil becomes magnetized to oppose the motion of the magnet. Work must be done against this repulsion to move the magnet toward the coil. This mechanical work is converted into electrical energy and stored in the coil.

Mathematical Application of Flux Change

• Calculation of Induced EMF: For a loop where the magnetic flux is given by the equation $\phi = 6t^2 + 7t + 1$ (where $t$ is in seconds and $\phi$ is in milliweber), the induced emf ($\epsilon$) is found by differentiating the flux with respect to time:     $\epsilon = \frac{d\phi}{dt} = 12t + 7$     To find the emf at $t = 2\,s$:     $\epsilon = 12(2) + 7 = 31\,mV$

Eddy Currents

• Definition: When the magnetic flux linked with a metallic block changes, an emf is induced within it. This emf produces circulating currents throughout the block, known as eddy currents. The direction of these currents is also determined by Lenz's law.

• Disadvantages of Eddy Currents: They are generally undesirable in electrical devices such as transformers, induction coils, and choke coils. They cause significant heating, which represents a wastage of energy.

• Minimization of Eddy Currents: These currents can be minimized by increasing the electrical resistance of the metal. This is achieved by using thin, laminated sheets of metal separated by an insulating layer, rather than using a single solid metallic block.

• Practical Applications of Eddy Currents:

  • Electromagnetic Damping: Used in moving coil galvanometers. As the coil oscillates within a magnetic field, eddy currents are set up in the frame (core) according to Lenz's law. These currents oppose the oscillations, providing quick damping for rapid measurements.

  • Induction Furnace: A metal to be melted is placed in a rapidly varying magnetic field produced by high-frequency alternating current (AC). The resulting eddy currents generate enough heat to melt the metal.

  • Electric Brakes (Train Brakes): A strong magnetic field is applied across a metallic drum rotating with the train's axis. The force developed by the eddy currents is proportional to the train's speed, ensuring smooth braking.

  • Induction Motor: A rotating magnetic field is produced using two single-phase AC currents with a phase difference of $\frac{\pi}{2}$. A metallic cylinder pivoted between electromagnets develops eddy currents that reduce relative motion, causing the cylinder to rotate with the field.

  • Speedometer: An aluminum drum rotates according to the vehicle's speed. As the vehicle moves, a magnet inside the drum rotates, producing eddy currents. These currents attempt to reduce relative motion, causing the drum to rotate with the magnet. A pointer indicates the speed based on this rotation.

Self-Induction and Inductance

• Definition of Self-Induction: When the current through a coil changes, an emf is induced within that same coil. This phenomenon is called self-induction.

• Self-Inductance (L): The magnetic flux $\phi$ is proportional to the current $I$:     $\phi \propto I \implies \phi = LI$     where $L$ is the coefficient of self-induction or self-inductance.

• Formula for Self-Induced EMF: According to Lenz’s law:     $\epsilon = -\frac{d\phi}{dt} = -\frac{d(LI)}{dt}$     $\epsilon = -L\frac{dI}{dt}$

• Defining Self-Inductance per Unit Rate of Change: Self-inductance is numerically equal to the induced emf developed in the coil when the rate of change of current through it is unity ($\frac{dI}{dt} = 1\,A/s$).

• Back EMF: The induced emf in a coil is called back emf because it opposes any growth or decay of the current passing through it.

• Units: The S.I. unit of self-inductance is the henry ($H$).

• Self-Inductance of a Solenoid: For a solenoid with total turns $N$, length $l$, and turns per unit length $n$:

  1. Magnetic flux linked: $\phi = BAN$

  2. Magnetic field inside solenoid: $B = \mu_0 n I$

  3. Since $\phi = LI$, then $LI = (\mu_0 n I)AN$

  4. $L = \mu_0 nAN$

  5. Substituting $N = nl$: $L = \mu_0 nA(nl) = \mu_0 n^2 Al$

Mutual Induction

• Definition: Mutual induction is the production of an induced emf in a neighboring coil when the current through the primary coil changes.

• Mutual Inductance (M): The flux linked with the secondary coil ($\phi$) is proportional to the current in the primary coil ($I_1$):     $\phi \propto I_1 \implies \phi = MI_1$     The induced emf in the secondary is:     $\epsilon = -\frac{d\phi}{dt} = -M\frac{dI_1}{dt}$

• Defining Mutual Inductance: Mutual inductance of two coils is defined as the induced emf developed in the secondary coil when the rate of change of current through the primary is unity ($\frac{dI_1}{dt} = 1$).

• Relationship between Mutual and Self-Inductance: The relation is given by:     $M = k \sqrt{L_1 L_2}$     where $k$ is the coefficient of coupling.

  • Tight Coupling: $k = 1$. This occurs when coils are wound closely so almost all flux from the primary links with the secondary.

  • Loose Coupling: k < 1.

• Units and Derivation for Coaxial Solenoids: The S.I. unit is the henry ($H$). For two coaxial solenoids $S_1$ and $S_2$ of length $l$ and radii $r_1, r_2$:     $M_{12} = \mu_0 n_1 n_2 (\pi r_1^2) l$     This demonstrates that $M_{12} = M_{21} = M$.

Energy Stored in an Inductor

• Mechanism: As current increases, back emf opposes the growth. The voltage source must do work to establish the current. This work is stored as magnetic potential energy.

• Derivation:

  1. Work done in time $dt$: $dW = P dt = \epsilon I dt$

  2. By Kirchhoff’s voltage rule: $\epsilon = L\frac{dI}{dt}$

  3. $dW = (L\frac{dI}{dt}) I dt = L I dI$

  4. Total work for current $0$ to $I_0$: $W = \int_0^{I_0} L I dI = L [\frac{I^2}{2}]_0^{I_0} = \frac{1}{2} L I_0^2$

  5. Energy of the inductor: $U_B = \frac{1}{2} L I^2$

Physics of Motion in Magnetic Fields

• Magnetic Flux ($\phi$): Defined as $\phi = BA\cos(\theta)$, where $\theta$ is the angle between the magnetic field and the normal to the plane of the coil. It is a scalar quantity measured in Weber ($Wb$).

• Motional EMF: Induced in a conductor moving through a magnetic field. For a conductor $PQ$ of length $l$ moving at velocity $v$ perpendicular to field $B$:     $\phi = Blx$     $\epsilon = -\frac{d\phi}{dt} = -\frac{d(Blx)}{dt} = -Bl\frac{dx}{dt}$     Since $-\frac{dx}{dt} = v$, the motional emf is:     $\epsilon = Blv$

• Fleming's Right Hand Rule: Stretch the thumb, forefinger, and middle finger of the right hand mutually perpendicular.

  • Forefinger = Direction of magnetic field.

  • Thumb = Direction of motion of the conductor.

  • Middle finger = Direction of the induced emf.

Alternating Current (A.C.) Generators

• Principle: Works on the principle of electromagnetic induction, converting mechanical energy into electrical energy.

• Construction Components:

  1. Field magnet

  2. Armature

  3. Slip rings

  4. Brushes

• Working and Mathematical Representation: As the armature rotates with angular velocity $\omega$, the flux changes as $\phi = BAN\cos(\omega t)$. The induced emf is:     $\epsilon = -\frac{d\phi}{dt} = -\frac{d}{dt}(NAB\cos(\omega t)) = NAB\omega \sin(\omega t)$     Maximum emf ($\epsilon_0$) occurs when $\sin(\omega t) = \pm 1$:     $\epsilon_{max} = NAB\omega$

• Commercial Advantages of AC: AC voltage is easily transformed using step-up and step-down transformers. Transmission at high voltage reduces $I^2R$ power losses in lines.

Questions & Discussion

• Current in wire AB is increasing; direct of induced current in adjacent loop?: According to Lenz's law, the induced current in the loop will flow in a direction that creates a magnetic field opposing the increase in the field from wire AB.

• How much emf is induced at t=2s for $\phi = 6t^2 + 7t + 1\,mWb$?: As derived, the induced emf is $31\,mV$.

• Current falls from 5A to 1A in 0.1s with 200V average emf; find inductance:     $\epsilon = -L \frac{\Delta I}{\Delta t} \implies 200 = -L \frac{1-5}{0.1} = L \frac{4}{0.1} = 40L \implies L = 5\,H$.

• Comparison of Inductors A and B on a $\phi$ vs $I$ plot: The inductor with the steeper slope has the larger value of self-inductance ($L = \phi / I$).

• Maximum induced EMF from 50A DC, 50A 50Hz AC, 50A 500Hz AC, or 100A DC?: 50A 500Hz AC produces the maximum induced emf because emf is proportional to the rate of change of current, which is highest at the highest frequency.

• Aluminum ring on an electromagnet core: When the circuit is closed, the increasing magnetic flux induces a current in the ring. According to Lenz's law, the ring develops a magnetic field that opposes the electromagnet's field, causing a repulsive force that makes the ring jump.

• Damping of oscillating copper plates: If slots are cut in the copper plates, the area available for eddy currents to circulate is reduced. This increases the resistance to the flow of eddy currents, thereby reducing the damping force and allowing the plate to oscillate longer.

• Jet plane voltage difference: A jet plane traveling west at $1800\,km/h$ ($500\,m/s$) with a wing span of $25\,m$ in Earth's field of $5 \times 10^{-4}\,T$ at a dip angle of $30^\circ$. The vertical component of the earth's field ($B_v = B\sin(30^\circ)$) is cut by the wings. The voltage is $\epsilon = B_v l v = (5 \times 10^{-4} \times 0.5) \times 25 \times 500 = 3.125\,V$.

Device Principles Match

• AC Generator: Electromagnetic Induction

• Choke Coil: Self-Induction

• Transformer: Mutual Induction

• Speedometer of Vehicles: Eddy Currents