Vardaan Learning Institute
Chapter 6: Electromagnetic Induction — Class Notes
1. Magnetic Flux ($\Phi_B$)
Definition: The total number of magnetic field lines passing normally (perpendicularly) through a given surface area $\vec{A}$ placed in a magnetic field $\vec{B}$.
Figure 6.4 & 6.5: Magnetic flux through planar surface and area elements \(d\vec{A}_i\) in magnetic field.
Formula & Properties
$$\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta = \int \vec{B} \cdot d\vec{A}$$
- $\theta$: Angle between magnetic field $\vec{B}$ and the normal / Area Vector $\vec{A}$.
- Quantity Type: Scalar quantity (can be positive, zero, or negative).
- SI Unit: Weber ($\text{Wb}$) or $\text{T}\cdot\text{m}^2 = \text{V}\cdot\text{s}$. CGS Unit: Maxwell ($1\text{ Wb} = 10^8\text{ Mx}$).
- Dimensional Formula: $[\text{M}\text{L}^2\text{T}^{-2}\text{A}^{-1}]$.
- Angular Conditions:
- Maximum Flux ($\Phi_{\max} = +BA$): $\theta = 0^\circ$ ($\vec{B} \parallel \vec{A}$, field lines normal to surface face).
- Zero Flux ($\Phi = 0$): $\theta = 90^\circ$ ($\vec{B} \perp \vec{A}$, field lines parallel/glancing to plane).
- Minimum Flux ($\Phi_{\min} = -BA$): $\theta = 180^\circ$ ($\vec{B}$ anti-parallel to $\vec{A}$).
2. Faraday's Experiments & Quantitative Laws of EMI
A. Landmark Experiments
- Experiment 6.1 (Magnet-Coil): Relative motion between a bar magnet and a coil induces an electric current in the coil. Faster motion produces larger deflection in the galvanometer.
- Experiment 6.2 (Coil-Coil): Relative motion between a primary current-carrying coil and a secondary coil induces an electric current in the secondary coil.
- Experiment 6.3 (Make & Break Current): No relative motion needed. A momentary induced current is produced during "make" (closing key) and "break" (opening key) of current in the neighboring primary coil.
Figure 6.1: Current induced in a coil due to relative motion of a bar magnet.
Figure 6.3: Current induced in secondary coil due to make and break of current in primary coil.
Faraday's Laws of Electromagnetic Induction
1. First Law (Qualitative): Whenever the magnetic flux linked with a closed circuit changes with time, an electromotive force (EMF) is induced in the circuit. The induced EMF lasts only as long as the magnetic flux change continues.
2. Second Law (Quantitative): The magnitude of the induced EMF is directly proportional to the time rate of change of magnetic flux linked with the circuit:
$$\epsilon = -N \frac{d\Phi_B}{dt}$$
(The negative sign represents the direction of induced EMF according to Lenz's Law).
Key Exam Relations: Induced Current & Charge
- Induced Current:
$$I = \frac{|\epsilon|}{R} = \frac{N}{R} \left|\frac{d\Phi_B}{dt}\right|$$
- Total Induced Charge ($\Delta q$):
$$\Delta q = I \Delta t = \frac{|\Delta\Phi_{\text{net}}|}{R}$$
- Vital Board Concept: Induced EMF and induced current depend strictly on the rate of change of flux (speed of motion $\Delta t$), whereas total induced charge $\Delta q$ is independent of time and speed.
3. Lenz's Law & Conservation of Energy
Statement: The polarity of the induced EMF is always such that it tends to produce an electric current which opposes the change in magnetic flux that produced it.
Figure 6.6: Direction of induced current opposing magnet motion.
End-Face Direction Rule & Energy Conservation
- Approaching North Pole: Flux $\uparrow \implies$ Facing loop develops North Polarity ($\circlearrowleft$ Anti-Clockwise Current) $\implies$ Repulsive force.
- Receding North Pole: Flux $\downarrow \implies$ Facing loop develops South Polarity ($\circlearrowright$ Clockwise Current) $\implies$ Attractive force.
- Approaching South Pole: Flux $\uparrow \implies$ Facing loop develops South Polarity ($\circlearrowright$ Clockwise Current) $\implies$ Repulsive force.
- Receding South Pole: Flux $\downarrow \implies$ Facing loop develops North Polarity ($\circlearrowleft$ Anti-Clockwise Current) $\implies$ Attractive force.
Conservation of Energy Justification: The mechanical work done by an external agent against the opposing magnetic force is transformed directly into electrical energy, which dissipates as Joule heat ($I^2Rt$). If Lenz's law were not true, an approaching magnet would experience attraction, gaining kinetic energy continuously without external work, violating the Law of Conservation of Energy.
Figure 6.7: Direction of induced current in planar loops entering and leaving magnetic field.
Figure 6.8 & 6.9: (c) Rectangular vs circular loop exiting field, and (d) Capacitor connected to loop with approaching magnets.
Standard Board Conceptual Cases
- Entering / Leaving Loops: Rectangular loop exiting at constant speed $v$ gives constant EMF ($\frac{dA}{dt} = lv = \text{const}$), while a circular loop gives a variable EMF because its cutting boundary width changes continuously.
- Capacitor Connected to Loop: When North poles approach from both sides, induced current charges plate A as Positive ($+$) and plate B as Negative ($-$).
4. Motional Electromotive Force
A. Translational Motional EMF
A straight conducting rod $PQ$ of length $l$ sliding with constant velocity $v$ perpendicular to a uniform magnetic field $\vec{B}$ directed into the page ($\otimes$).
Figure 6.10: Arm PQ sliding across U-shaped rails in uniform magnetic field.
Board Derivation: Translational Motional EMF
1. Flux Rule Method: Area enclosed by loop $PQRS$ at instant $t$ is $A = lx$. Flux linked $\Phi_B = BA = Blx$.
$$\epsilon = -\frac{d\Phi_B}{dt} = -Bl\left(\frac{dx}{dt}\right) = -Bl(-v) \implies \mathbf{\epsilon = Blv}$$
2. Lorentz Force Method: Magnetic force on free electron/charge $q$ is $F_m = qvB$. Work done across length $l$ is $W = F_m \cdot l = qvBl$.
$$\epsilon = \frac{W}{q} = \mathbf{Blv}$$
Power & Energy Balance in Motional EMF
- Induced Current: $I = \frac{\epsilon}{R} = \frac{Blv}{R}$
- Retarding Magnetic Force: $F_{\text{mag}} = IlB = \frac{B^2 l^2 v}{R}$
- Mechanical Power Supplied: $P_{\text{mech}} = F_{\text{ext}} \cdot v = \frac{B^2 l^2 v^2}{R}$
- Joule Heating Rate: $P_{\text{Joule}} = I^2 R = \frac{B^2 l^2 v^2}{R} \implies \mathbf{P_{\text{mech}} = P_{\text{Joule}}}$
B. Rotational Motional EMF
A conducting rod of length $l$ (or disc of radius $R$) rotating about one end with angular velocity $\omega$ (frequency $\nu$) in a uniform perpendicular magnetic field $\vec{B}$.
Figure 6.11: Rotating metallic rod hinged at center touching circular metallic ring.
Board Derivation: Rotational Motional EMF
For an infinitesimal element $dr$ at distance $r$ from the axle, linear velocity is $v = \omega r$. Elementary EMF:
$$d\epsilon = B v dr = B(\omega r) dr = B\omega r dr$$
Integrating from $r = 0$ to $r = l$:
$$\epsilon = \int_0^l B\omega r dr = B\omega \left[\frac{r^2}{2}\right]_0^l = \mathbf{\frac{1}{2} B \omega l^2} = \mathbf{B A \nu} \quad (\text{where } A = \pi l^2, \omega = 2\pi\nu)$$
Wheel with $N$ Spokes: All $N$ spokes are connected in parallel between axle and rim. Total EMF remains equal to EMF of a single spoke: $\epsilon = \frac{1}{2}B\omega R^2$.
5. Eddy Currents (Foucault Currents)
Definition: Circulating induced electric currents produced in the bulk/body of a conductor when the magnetic flux linked with it changes with time.
Key Board Aspects: Eddy Currents
- Undesirable Effect: Cause substantial energy loss in the form of heat in the soft iron cores of transformers, induction coils, and dynamos.
- Reduction Technique: Using a laminated core composed of thin sheets insulated from one another by lacquer or varnish (breaks eddy current loops, increasing electrical resistance).
- Practical Applications (Frequently Asked in Boards):
- Magnetic Braking in Trains: Electromagnets induce eddy currents in rails/wheels, producing opposing torque that smoothly brakes the train without mechanical wear.
- Electromagnetic Damping (Dead-Beat Galvanometers): Coil wound on metallic frame produces eddy currents that oppose oscillation and bring pointer to rest quickly.
- Induction Furnace: High-frequency changing field induces powerful eddy currents that melt metals cleanly in vacuum.
- Electric Power Meters: Rotating shiny aluminium disc powered by eddy currents.
6. Self-Inductance & Stored Magnetic Energy
Definition: The property of an electric circuit by virtue of which it opposes any change in the current flowing through itself by inducing an opposing Back EMF (Electrical Inertia).
$$N\Phi_B = L I \implies \epsilon = -L \frac{dI}{dt}$$
Where $L$ is the Coefficient of Self-Inductance (SI Unit: Henry, $\text{H} = \text{Wb/A} = \text{V}\cdot\text{s/A} = \Omega\cdot\text{s}$).
Board Derivation: Self-Inductance of a Long Solenoid
For a long solenoid of length $l$, cross-sectional area $A$, turn density $n = \frac{N}{l}$ carrying current $I$:
- Internal magnetic field: $B = \mu_0 n I = \mu_0 \left(\frac{N}{l}\right) I$
- Flux linked per turn: $\phi = BA = \mu_0 \left(\frac{N}{l}\right) I A$
- Total flux linkage: $N\Phi_B = N \phi = \frac{\mu_0 N^2 A}{l} I = (\mu_0 n^2 A l) I$
$$\mathbf{L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 A l = \mu_r \mu_0 n^2 A l}$$
Stored Magnetic Energy & Energy Density
- Work Done / Magnetic Energy:
$$dW = |\epsilon| I dt = L I dI \implies \mathbf{U_B = \int_0^I L I dI = \frac{1}{2} L I^2}$$
- Magnetic Energy Density ($u_B$):
$$u_B = \frac{U_B}{\text{Volume}} = \mathbf{\frac{B^2}{2\mu_0}} \quad (\text{J/m}^3)$$
(Analogous to electrostatic energy density in a capacitor: $u_E = \frac{1}{2}\epsilon_0 E^2$).
- Combination of Inductors:
- Series: $L_s = L_1 + L_2$
- Parallel: $\frac{1}{L_p} = \frac{1}{L_1} + \frac{1}{L_2}$
7. Mutual Induction
Definition: The phenomenon of inducing an EMF in a secondary coil due to a changing current in a neighboring primary coil.
$$N_1 \Phi_1 = M_{12} I_2 \implies \epsilon_1 = -M_{12} \frac{dI_2}{dt}$$
Figure 6.12: Two long co-axial solenoids \(S_1\) and \(S_2\) of length \(l\).
Board Derivation: Mutual Inductance of Two Coaxial Solenoids
Consider two co-axial solenoids of length $l$ ($r_1 < r_2$):
- Current $I_2$ in outer solenoid $S_2$ produces field $B_2 = \mu_0 n_2 I_2$.
- Flux linked with inner solenoid $S_1$ (confined to area $A_1 = \pi r_1^2$):
$$N_1 \Phi_1 = (n_1 l)(\pi r_1^2)(\mu_0 n_2 I_2) = (\mu_0 n_1 n_2 \pi r_1^2 l) I_2$$
$$\mathbf{M_{12} = M_{21} = M = \mu_0 n_1 n_2 \pi r_1^2 l = \frac{\mu_0 N_1 N_2 A_1}{l} = \mu_r \frac{\mu_0 N_1 N_2 A_1}{l}}$$
Concentric Co-planar Coils ($r_1 \ll r_2$): $M = \mathbf{\frac{\mu_0 \pi r_1^2}{2 r_2}}$
Coupling Coefficient ($K$): $K = \frac{M}{\sqrt{L_1 L_2}}$ ($0 \le K \le 1$). $K = 1$ for perfect flux linkage, $K = 0$ when perpendicular.
8. AC Generator (Alternator)
Principle: Converts mechanical energy into alternating electrical energy using electromagnetic induction by rotating an armature coil in a uniform magnetic field.
Figure 6.13: Schematic diagram of an AC Generator mechanism.
Figure 6.14: Stages of armature coil rotation and generated alternating EMF.
Board Derivation: Mathematical Theory of AC Generator
For an armature coil of $N$ turns, area $A$ rotating with uniform angular velocity $\omega$ about an axis perpendicular to field $\vec{B}$:
- Angle of rotation at time $t$: $\theta = \omega t$
- Magnetic flux linkage: $\Phi_{\text{total}} = N B A \cos(\omega t)$
- Induced EMF by Faraday's Law:
$$\epsilon = -\frac{d\Phi_{\text{total}}}{dt} = -NBA \frac{d}{dt}[\cos(\omega t)] = NBA\omega \sin(\omega t)$$
$$\mathbf{\epsilon = \epsilon_0 \sin(\omega t) = \epsilon_0 \sin(2\pi \nu t)}$$
Where Peak EMF is $\mathbf{\epsilon_0 = NBA\omega = 2\pi \nu NBA}$.
Alternating Current: $\mathbf{I = I_0 \sin(\omega t)}$ where $I_0 = \frac{\epsilon_0}{R}$.
Rotation Stages & Generated Waveform
| Time ($t$) |
Angle ($\theta = \omega t$) |
Plane of Coil relative to $\vec{B}$ |
Flux Linkage ($\Phi$) |
Induced EMF ($\epsilon$) |
| $0$ |
$0^\circ$ |
Perpendicular to $\vec{B}$ |
Maximum ($+NBA$) |
$\mathbf{0}$ |
| $T/4$ |
$90^\circ$ |
Parallel to $\vec{B}$ |
Zero ($0$) |
$\mathbf{+\epsilon_0}$ (Positive Peak) |
| $T/2$ |
$180^\circ$ |
Perpendicular to $\vec{B}$ (inverted) |
Maximum ($-NBA$) |
$\mathbf{0}$ |
| $3T/4$ |
$270^\circ$ |
Parallel to $\vec{B}$ |
Zero ($0$) |
$\mathbf{-\epsilon_0}$ (Negative Peak) |
| $T$ |
$360^\circ$ |
Perpendicular to $\vec{B}$ (initial) |
Maximum ($+NBA$) |
$\mathbf{0}$ |
9. Comprehensive Formula Revision Table
| Physical Quantity |
Symbol |
Core Formula |
SI Unit |
Dimensions |
| Magnetic Flux |
$\Phi_B$ |
$\Phi_B = BA\cos\theta = \vec{B}\cdot\vec{A}$ |
Weber ($\text{Wb}$) |
$[\text{M}\text{L}^2\text{T}^{-2}\text{A}^{-1}]$ |
| Faraday's Induced EMF |
$\epsilon$ |
$\epsilon = -N \frac{d\Phi_B}{dt}$ |
Volt ($\text{V}$) |
$[\text{M}\text{L}^2\text{T}^{-3}\text{A}^{-1}]$ |
| Induced Charge |
$\Delta q$ |
$\Delta q = \frac{|\Delta\Phi|}{R}$ |
Coulomb ($\text{C}$) |
$[\text{A}\text{T}]$ |
| Translational Motional EMF |
$\epsilon$ |
$\epsilon = Blv$ |
Volt ($\text{V}$) |
$[\text{M}\text{L}^2\text{T}^{-3}\text{A}^{-1}]$ |
| Rotational Motional EMF |
$\epsilon$ |
$\epsilon = \frac{1}{2}B\omega l^2 = BA\nu$ |
Volt ($\text{V}$) |
$[\text{M}\text{L}^2\text{T}^{-3}\text{A}^{-1}]$ |
| Self-Inductance (Solenoid) |
$L$ |
$L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 Al$ |
Henry ($\text{H}$) |
$[\text{M}\text{L}^2\text{T}^{-2}\text{A}^{-2}]$ |
| Mutual Inductance (Coaxial) |
$M$ |
$M = \mu_0 n_1 n_2 \pi r_1^2 l = \frac{\mu_0 N_1 N_2 A_1}{l}$ |
Henry ($\text{H}$) |
$[\text{M}\text{L}^2\text{T}^{-2}\text{A}^{-2}]$ |
| Magnetic Energy Stored |
$U_B$ |
$U_B = \frac{1}{2} L I^2$ |
Joule ($\text{J}$) |
$[\text{M}\text{L}^2\text{T}^{-2}]$ |
| Magnetic Energy Density |
$u_B$ |
$u_B = \frac{B^2}{2\mu_0}$ |
$\text{J/m}^3$ |
$[\text{M}\text{L}^{-1}\text{T}^{-2}]$ |
| AC Generator Peak EMF |
$\epsilon_0$ |
$\epsilon_0 = NBA\omega = 2\pi\nu NBA$ |
Volt ($\text{V}$) |
$[\text{M}\text{L}^2\text{T}^{-3}\text{A}^{-1}]$ |
