EMF Laws and Induction Notes for GATE exam

Faraday’s Law

Faraday’s Law - Fundamental Statement

  • Statement: The induced EMF in a closed loop equals the negative rate of change of magnetic flux through the loop.

  • Mathematical Form:

    \[\boxed{\mathcal{E} = -\frac{d}{dt} \Phi_B} \quad \text{where} \quad \Phi_B = \int \mathbf{B} \cdot d\mathbf{A}\]
  • Alternative Forms:

    \[\begin{aligned} \mathcal{E} &= -\frac{d}{dt} \int \mathbf{B} \cdot d\mathbf{A} \\ \oint \mathbf{E} \cdot d\mathbf{l} &= -\frac{d}{dt} \int \mathbf{B} \cdot d\mathbf{A} \end{aligned}\]
  • Lenz’s Law: The negative sign indicates that induced current opposes the change in flux.

Faraday’s Law - Key Parameters

  • Magnetic Flux (\(\Phi_B\)):

    • Unit: Weber (Wb) = T m2

    • \(\Phi_B = B \cdot A \cdot \cos\theta\) (for uniform field)

    • \(\theta\): Angle between \(\mathbf{B}\) and area normal

  • Induced EMF (\(\mathcal{E}\)):

    • Unit: Volt (V)

    • Direction determined by Lenz’s Law

  • Three ways to change flux:

    1. Change magnetic field strength (\(B\))

    2. Change loop area (\(A\))

    3. Change orientation (\(\theta\))

Faraday’s Law - GATE Applications

  • Moving Conductor in Magnetic Field:

    \[\mathcal{E} = B \cdot l \cdot v \cdot \sin\theta\]
    where \(l\) = length, \(v\) = velocity, \(\theta\) = angle between \(\mathbf{v}\) and \(\mathbf{B}\)
  • Rotating Coil in Magnetic Field:

    \[\mathcal{E} = NBA\omega \sin(\omega t)\]
    where \(N\) = number of turns, \(\omega\) = angular frequency
  • Transformers: \(\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{N_1}{N_2}\)

  • GATE Tip: Always check the direction of induced current using right-hand rule!

Lorentz Force

Lorentz Force - Complete Expression

  • Definition: Total force on a charged particle in electric and magnetic fields.

  • Complete Formula:

    \[\boxed{\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})}\]
  • Component Analysis:

    • Electric Force: \(\mathbf{F}_E = q\mathbf{E}\) (parallel to \(\mathbf{E}\))

    • Magnetic Force: \(\mathbf{F}_B = q\mathbf{v} \times \mathbf{B}\) (perpendicular to both \(\mathbf{v}\) and \(\mathbf{B}\))

  • Magnitude of Magnetic Force:

    \[|\mathbf{F}_B| = |q|vB\sin\theta\]
    where \(\theta\) is angle between \(\mathbf{v}\) and \(\mathbf{B}\)

Lorentz Force - Applications

  • Cyclotron Motion:

    • Radius: \(r = \frac{mv}{qB}\)

    • Frequency: \(f = \frac{qB}{2\pi m}\) (independent of \(v\)!)

    • Period: \(T = \frac{2\pi m}{qB}\)

  • Velocity Selector: \(v = \frac{E}{B}\) (when \(\mathbf{E} \perp \mathbf{B}\))

  • Hall Effect:

    \[V_H = \frac{IB}{nqt} \quad \text{(Hall Voltage)}\]
    where \(n\) = charge density, \(t\) = thickness
  • GATE Focus: Direction problems using right-hand rule, cyclotron calculations

Lorentz Force - Problem Solving Tips

  • Direction Rules:

    • Right-hand rule: Point fingers in direction of \(\mathbf{v}\), curl towards \(\mathbf{B}\), thumb shows \(\mathbf{F}\) for positive charge

    • For negative charge: Reverse the direction

  • Special Cases:

    • \(\mathbf{v} \parallel \mathbf{B}\): No magnetic force (\(\sin\theta = 0\))

    • \(\mathbf{v} \perp \mathbf{B}\): Maximum force (\(\sin\theta = 1\))

    • Stationary charge: Only electric force

  • Energy Considerations:

    • Magnetic force does NO work (always \(\perp\) to motion)

    • Electric force can do work

Inductance

Inductance - Fundamental Concepts

  • Definition: Property of a conductor/circuit to oppose changes in current through induced EMF.

  • Physical Origin: Changing current creates changing magnetic field, which induces EMF (Faraday’s Law)

  • Unit: Henry (H)

    \[1 \text{ Henry} = 1 \frac{\text{Weber}}{\text{Ampere}} = 1 \frac{\text{Volt·second}}{\text{Ampere}}\]
  • Energy Storage:

    \[U = \frac{1}{2}LI^2 \quad \text{(Energy stored in inductor)}\]
  • Key Insight: Inductance depends only on geometry and material properties, not on current!

Types of Inductance

  • Self-Inductance (\(L\)):

    • EMF induced in same circuit due to its own changing current

    • \(\mathcal{E} = -L \frac{dI}{dt}\)

    • Always opposes change in current

  • Mutual Inductance (\(M\)):

    • EMF induced in one circuit due to changing current in another

    • \(\mathcal{E}_2 = -M \frac{dI_1}{dt}\)

    • Basis for transformers

  • Relationship:

    \[M_{12} = M_{21} = M \quad \text{(Reciprocity)}\]
  • GATE Note: Both self and mutual inductance can be positive or negative depending on winding direction!

Self and Mutual Inductance

Self-Inductance - Mathematical Relations

  • General Definition:

    \[L = \frac{\Phi_B}{I} = \frac{N\Phi_B}{I}\]
    where \(N\) = number of turns
  • Induced EMF:

    \[\mathcal{E} = -L \frac{dI}{dt}\]
  • From Magnetic Field Energy:

    \[L = \frac{2U_B}{I^2} \quad \text{where} \quad U_B = \frac{B^2}{2\mu_0} \times \text{Volume}\]
  • Important: Self-inductance is always positive in our sign convention

Self-Inductance - Common Configurations

  • Solenoid (Long):

    \[\boxed{L = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}}\]
    where \(n = N/l\) (turns per unit length)
  • Toroidal Coil:

    \[L = \frac{\mu_0 N^2 A}{2\pi r}\]
    where \(r\) = mean radius of torus
  • Single-turn Circular Loop:

    \[L = \mu_0 R \left[\ln\left(\frac{8R}{a}\right) - 2\right]\]
    where \(R\) = loop radius, \(a\) = wire radius
  • GATE Focus: Solenoid formula is most frequently asked!

Mutual Inductance - Mathematical Relations

  • Definition:

    \[M_{21} = \frac{\Phi_{21}}{I_1} = \frac{N_2\Phi_{21}}{I_1}\]
  • Induced EMF:

    \[\mathcal{E}_2 = -M \frac{dI_1}{dt}\]
  • Coupling Coefficient:

    \[k = \frac{M}{\sqrt{L_1 L_2}} \quad \text{where} \quad 0 \leq k \leq 1\]
    • \(k = 0\): No coupling

    • \(k = 1\): Perfect coupling

  • Maximum Mutual Inductance:

    \[M_{\text{max}} = \sqrt{L_1 L_2}\]

Mutual Inductance - Common Configurations

  • Coaxial Solenoids:

    \[M = \frac{\mu_0 N_1 N_2 A}{l} \quad \text{(inner solenoid area $A$)}\]
  • Two Parallel Coaxial Circular Loops:

    \[M = \frac{\mu_0 \sqrt{R_1 R_2}}{2} \left[(2-k^2)K(k) - 2E(k)\right]\]
    where \(K(k)\) and \(E(k)\) are elliptic integrals
  • Transformer (Ideal):

    \[M = k\sqrt{L_1 L_2} \quad \text{with} \quad k \approx 1\]
  • GATE Tip: For most problems, use coaxial solenoid formula!

Inductance - Circuit Analysis

  • Series Connection:

    \[L_{\text{total}} = L_1 + L_2 + 2M \quad \text{(aiding)}\]
    \[L_{\text{total}} = L_1 + L_2 - 2M \quad \text{(opposing)}\]
  • Parallel Connection:

    \[\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} \quad \text{(when $M = 0$)}\]
  • RL Circuit Time Constant:

    \[\tau = \frac{L}{R} \quad \text{and} \quad I(t) = I_0\left(1 - e^{-t/\tau}\right)\]
  • Important: Sign of mutual inductance depends on relative winding directions!

GATE Problem-Solving Strategy

  • For Faraday’s Law Problems:

    • Identify what’s changing (B, A, or \(\theta\))

    • Apply \(\mathcal{E} = -\frac{d\Phi_B}{dt}\)

    • Use Lenz’s Law for direction

  • For Lorentz Force Problems:

    • Identify all fields present

    • Apply right-hand rule for direction

    • Calculate magnitude using \(F = qvB\sin\theta\)

  • For Inductance Problems:

    • Identify geometry (solenoid, toroidal, etc.)

    • Apply appropriate formula

    • Check units and reasonableness

Key Formulas Summary

  • Faraday’s Law: \(\mathcal{E} = -\frac{d\Phi_B}{dt}\)

  • Lorentz Force: \(\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\)

  • Cyclotron Frequency: \(f = \frac{qB}{2\pi m}\)

  • Solenoid Self-Inductance: \(L = \frac{\mu_0 N^2 A}{l}\)

  • Mutual Inductance: \(M = \frac{\mu_0 N_1 N_2 A}{l}\) (coaxial solenoids)

  • Coupling Coefficient: \(k = \frac{M}{\sqrt{L_1 L_2}}\)

  • Energy in Inductor: \(U = \frac{1}{2}LI^2\)

  • RL Time Constant: \(\tau = \frac{L}{R}\)

Summary and GATE Tips

  • Master the Basics:

    • Faraday’s Law: Focus on flux change calculations

    • Lorentz Force: Practice direction problems

    • Inductance: Memorize solenoid formulas

  • Common GATE Topics:

    • EMF in moving conductors

    • Cyclotron motion calculations

    • Transformer principles

    • Mutual inductance of coaxial coils

  • Problem-Solving Tips:

    • Always check units in final answers

    • Use symmetry to simplify calculations

    • Draw clear diagrams for direction problems