Class 12th Physics - Electromagnetic Induction (Chapter 6)

Introduction to Electromagnetic Induction

  • Definition of Electromagnetic Induction: The phenomenon of inducing an electromotive force (emf) by changing the magnetic flux (i.e., the number of magnetic field lines) passing through a coil is known as electromagnetic induction.
  • Induced Current: If the electric circuit containing the coil is closed, the induced electromotive force causes an electric current to flow through the circuit. This current is termed the induced current.
  • Fundamental Components:
    • A magnetic field \\\vec{B}\\\).\n * Magnetic field lines of force.\n * A coil/circuit exposed to the magnetic field.\n * Absence of current when magnetic flux remains constant; current is generated only when flux changes.\n\n# Experiments of Faraday and Henry\n\n* **Experiment 1: Current Induced by a Magnet**\n * **Motion Towards Coil:** When a magnet moves toward a stationary coil C_1, the number of magnetic field lines passing through the coil increases (changes). This change in magnetic flux induces an electric current in the coil.\n * **Motion Away from Coil:** When the magnet moves away from the coil C_1, the number of magnetic field lines passing through the coil decreases (changes). Current is again induced in the coil, but its direction is opposite to that observed when moving toward the coil.\n * **Core Conclusion:** An induced current is generated in the circuit **only when there is relative motion** between the coil and the magnet.\n\n* **Experiment 2: Current Induced by Current**\n * **Primary and Secondary Coil Interaction:** When a primary coil carrying an electric current is moved toward a secondary stationary coil, the magnetic field lines crossing the secondary coil increase (change).\n * **Induced Effect:** This change in magnetic flux induces an electric current in the secondary coil without any direct physical contact between the two coils.\n\n* **Experiment 3: Current Induced by Changing Current**\n * **Switching ON the Circuit:** When the primary circuit is switched ON, the current builds up from zero to a maximum value, causing a rapid change in magnetic flux. This produces a **momentary deflection** in the Galvanometer connected to the secondary coil.\n * **Steady Current State:** While the current remains constant after switching ON, the magnetic flux does not change, and the Galvanometer deflection drops to zero.\n * **Switching OFF the Circuit:** When the primary circuit is switched OFF, the current collapses to zero, producing another sudden change in magnetic flux. This causes a **momentary deflection in the Galvanometer in the opposite direction**.\n\n# Magnetic Flux\n\n* **Definition:** The magnetic flux (denoted by \Phi)throughanysurfaceofarea) through any surface of areaAplacedinamagneticfieldplaced in a magnetic field\vec{B} is measured by the total number of magnetic field lines of force crossing that surface normally.\n\n* **Mathematical Derivation:**\n * Let \thetabetheanglebetweentheuniformmagneticfieldbe the angle between the uniform magnetic field\vec{B}andthenormaltothesurfaceareavectorand the normal to the surface area vector\vec{A}.\n * The effective area perpendicular to the magnetic field direction is A \cos(\theta).\n * Thus, magnetic flux is given by:\n    \Phi = B A \cos(\theta)\n * Expressed as a vector dot product:\n    \Phi = \vec{B} \cdot \vec{A}\n\n* **Magnetic Flux in a Variable Magnetic Field:**\n * When the magnetic field is non-uniform across a surface, the total area is divided into nsmallinfinitesimalareaelementssmall infinitesimal area elementsd\vec{A}, such that the magnetic field over each tiny element can be treated as uniform.\n * Flux through individual elements:\n    \Phi_1 = \vec{B}_1 \cdot \vec{A}_1\n    \Phi_2 = \vec{B}_2 \cdot \vec{A}_2\n    \Phi_3 = \vec{B}_3 \cdot \vec{A}_3\n    \Phi_n = \vec{B}_n \cdot \vec{A}_n\n * Total magnetic flux is the sum of contributions from all surface elements:\n    \Phi = \vec{B}_1 \cdot \vec{A}_1 + \vec{B}_2 \cdot \vec{A}_2 + \dots + \vec{B}_n \cdot \vec{A}_n\n    \Phi = \sum \vec{B} \cdot \vec{A}\n * Continuous surface integral representation:\n    \Phi = \int \vec{B} \cdot d\vec{A}\n\n* **Special Cases:**\n * **Case 1 (Maximum Flux):** When the magnetic field is perpendicular to the surface area (\theta = 0^\circrelativetothenormalvectorrelative to the normal vector\vec{A}):\n    \Phi = B A \cos(0^\circ) = B A\n * **Case 2 (Zero Flux):** When the magnetic field is parallel to the surface plane (\theta = 90^\circrelativetothenormalvectorrelative to the normal vector\vec{A}):\n    \Phi = B A \cos(90^\circ) = 0\n\n* **Physical Nature and Units of Magnetic Flux:**\n * **Physical Quantity:** Magnetic flux is a **scalar quantity**.\n * **S.I. Unit:** Weber (\text{Wb}).\n * **C.G.S. Unit:** Maxwell (\text{Mx}).\n * **Unit Conversion:**\n    1\,\text{Wb} = 10^8\,\text{Maxwell}\n\n# Faraday's Laws of Electromagnetic Induction\n\n* **Faraday's First Law:**\n * Whenever the amount of magnetic flux linked with a circuit changes, an electromotive force (emf) is induced in the circuit.\n * The induced emf lasts for as long as the change in magnetic flux continues.\n\n* **Faraday's Second Law:**\n * The magnitude of emf induced in a circuit is directly proportional to the rate of change of magnetic flux linked with the circuit.\n * If \Phi_1istheinitialfluxlinkedwiththecoilatanyinstantandis the initial flux linked with the coil at any instant and\Phi_2isthemagneticfluxlinkedaftertimeintervalis the magnetic flux linked after time interval\Delta t:\n    e \propto \frac{\Phi_2 - \Phi_1}{\Delta t}\n    e = k \left( \frac{\Phi_2 - \Phi_1}{\Delta t} \right)\n * Setting proportionality constant k = 1:\n    e = \frac{\Phi_2 - \Phi_1}{\Delta t} = \frac{\Delta \Phi}{\Delta t}\n * For an infinitesimal time interval dt, the induced emf is expressed in differential form as:\n    e = -\frac{d\Phi}{dt}\n * The negative sign signifies that the induced emf opposes the change in magnetic flux responsible for its generation.\n\n* **EMF for a Coil with N Turns:**\n * For a coil containing N turns, the total induced emf is:\n    e = -N \frac{d\Phi}{dt}\n\n* **Induced Current Formula:**\n * If the coil/circuit has a total electrical resistance R,theinducedcurrent, the induced currenti is:\n    i = \frac{e}{R} = -\frac{N}{R} \frac{d\Phi}{dt}\n\n* **Worked Example Problem:**\n * **Problem Statement:** A coil having 500squareloops,eachofsidesquare loops, each of side10\,\text{cm},isplacednormaltoamagneticfieldwhichincreasesattherateof, is placed normal to a magnetic field which increases at the rate of1.0\,\text{Tesla/sec}. What is the e.m.f. induced in the loop?\n * **Given Data:**\n * Number of turns, N = 500\n * Side of square loop, s = 10\,\text{cm} = 10 \times 10^{-2}\,\text{m} = 0.1\,\text{m}\n * Area of single loop, A = s^2 = (10 \times 10^{-2}\,\text{m})^2 = 10^{-2}\,\text{m}^2\n * Rate of change of magnetic field, \frac{dB}{dt} = 1.0\,\text{T/s}\n * Orientation angle relative to normal, \theta = 0^\circ(sincethesurfaceisnormaltothefieldlines,theareavectorisparalleltofieldlines,(since the surface is normal to the field lines, the area vector is parallel to field lines,\cos(0^\circ) = 1)\n * **Step-by-Step Solution:**\n    e = -N \frac{d\Phi}{dt}\n    e = -500 \frac{d}{dt} \left( B A \cos(0^\circ) \right)\n    e = -500 \frac{d}{dt} (B A)\n    e = -500 A \frac{dB}{dt}\n    e = -500 \times (10 \times 10^{-2})^2 \times 1\n    e = -500 \times 10^{-2}\n    e = -5\,\text{V}\n * **Final Answer:** The induced emf in the coil is -5\,\text{V}(magnitude(magnitude5\,\text{V}$$).

Lenz's Law and Energy Conservation

  • Lenz's Law Statement:

    • According to Lenz's law, the direction of induced current is always such that it opposes the change in magnetic flux responsible for its production.

  • Experimental Verification of Lenz's Law:

    • North Pole Approaching: When the North pole of a magnet is brought towards a coil, the facing side of the coil develops North polarity (anticlockwise current flow) to oppose the approach of the magnet.
    • Galvanometer Response: For an anticlockwise induced current, the deflection in the connected Galvanometer is towards the right.
    • South Pole Approaching / North Pole Receding: Corresponding opposite magnetic polarities are induced on the coil face to oppose the relative motion.

  • Energy Conservation in Lenz's Law:

    • When a magnet is moved against the repulsive or attractive magnetic forces induced in the coil, external mechanical work must be performed.
    • The mechanical work done in overcoming this opposition to change the magnetic flux through the coil is converted directly into electrical energy.
    • Hence, Lenz's law is a direct consequence of the Law of Conservation of Energy.