Chapter 15: Electromagnetic Induction

• Electromagnetic Induction (EMI)

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  • Phenomenon of producing an induced electromotive force (EMF) and hence an induced current in a conductor due to a change in magnetic flux linked with the conductor.

• Magnetic Flux ($\Phi_B$)

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  • Measure of the number of magnetic field lines passing through a given area.
  • $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$.
  • Unit: Weber (Wb).
  • Scalar quantity.

• Faraday's Law of Electromagnetic Induction

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  • Induced EMF is directly proportional to the negative rate of change of magnetic flux.
  • $\mathcal{E} = -N \frac{\Delta\Phi_B}{\Delta t}$.
  • For a single loop: $\mathcal{E} = -\frac{\Delta\Phi_B}{\Delta t}$.
  • $N$: number of turns in the coil.
  • Negative sign indicates direction of induced EMF (Lenz's Law).

• Lenz's Law

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  • The direction of the induced current (or EMF) is such that it opposes the cause producing it.
  • Based on the principle of conservation of energy.

• Motional EMF

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  • EMF induced in a conductor moving in a magnetic field.
  • $\mathcal{E} = (vBL \sin\theta)$.
  • $v$: velocity of conductor.
  • $B$: magnetic field strength.
  • $L$: length of conductor in the field.
  • $\theta$: angle between $\vec{v}$ and $\vec{B}$.
  • Maximum when $\vec{v}$ is perpendicular to $\vec{B}$ ($\mathcal{E} = vBL$).

• Self-Induction

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  • Phenomenon where a changing current in a coil induces an EMF in the same coil.
  • Induced EMF: $\mathcal{E}_L = -L \frac{\Delta I}{\Delta t}$.
  • $L$: self-inductance of the coil.

  • Self-Inductance (L)

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    • Property of a coil that opposes change in current through it.
    • Unit: Henry (H) = V s/A.
    • Dimensions: $[ML^2T^{-2}A^{-2}]$.
    • For a solenoid: $L = \frac{\mu_o N^2 A}{l}$.
    • $\mu_o$: permeability of free space.
    • $N$: number of turns.
    • $A$: cross-sectional area.
    • $l$: length of solenoid.

  • Energy Stored in an Inductor

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    • $U = \frac{1}{2}LI^2$.
    • Unit: Joule (J).

• Mutual Induction

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  • Phenomenon where a changing current in one coil (primary) induces an EMF in a neighboring coil (secondary).
  • Induced EMF in secondary: $\mathcal{E}_s = -M \frac{\Delta I_p}{\Delta t}$.
  • $M$: mutual inductance.

  • Mutual Inductance (M)

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    • Property of two coils that determines the EMF induced in one due to current change in the other.
    • Unit: Henry (H).
    • For two coaxial solenoids: $M = \frac{\mu_o N_p N_s A}{l}$.
    • $N_p, N_s$: turns in primary and secondary coils.
    • $A$: common cross-sectional area.
    • $l$: length of coils.

• AC Generator (Dynamo)

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  • Converts mechanical energy into electrical energy.
  • Principle: Electromagnetic induction.
  • Induced EMF: $\mathcal{E} = N A B \omega \sin(\omega t)$.
  • Maximum EMF: $\mathcal{E}_{max} = N A B \omega$.
  • Output is sinusoidal AC voltage.

• Transformer

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  • Device that changes AC voltage levels.
  • Principle: Mutual induction.
  • Consists of primary and secondary coils wound on a soft iron core.

  • Transformer Equation

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    • $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$.
    • $V_s, V_p$: secondary and primary voltages.
    • $N_s, N_p$: secondary and primary turns.
    • $I_s, I_p$: secondary and primary currents.

  • Types of Transformers

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    • Step-up Transformer: $N_s > N_p \implies V_s > V_p$ (and $I_s < I_p$).
    • Step-down Transformer: $N_s < N_p \implies V_s < V_p$ (and $I_s > I_p$).

  • Efficiency of Transformer

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    • $\eta = \frac{\text{Output Power}}{\text{Input Power}} \times 100\% = \frac{V_s I_s}{V_p I_p} \times 100\%$.
    • Ideal transformer: $\eta = 100\%$.
    • Energy losses due to: flux leakage, resistance of windings (copper loss), eddy currents, hysteresis loss.

• Eddy Currents

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  • Circulating currents induced in bulk conductors when exposed to changing magnetic flux.
  • Produce heat (energy loss).
  • Minimized by laminating transformer cores.
  • Applications: Induction furnaces, electromagnetic braking.

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