Electromagnetic Induction & Alternating Currents Summary

Magnetic Flux, Faraday's Law, Inductance, AC Circuits, and Transformers

Magnetic Flux
Faraday's Law
Inductance
AC Circuits
Transformers

1. Magnetic Flux

Magnetic flux (ΦB) through an area dS in a magnetic field B is defined as the surface integral of the magnetic field over that area.

\[ \Phi_B = \int \vec{B} \cdot d\vec{S} \]

For an elemental area dS in a magnetic field B, the associated magnetic flux is:

\[ d\Phi_B = \vec{B} \cdot d\vec{S} = B \, dS \cos \theta \]
Gauss' Law for Magnetism

Since magnetic field lines are closed curves (no magnetic monopoles exist), the total magnetic flux linked with a closed surface is always zero:

\[ \oint \vec{B} \cdot d\vec{S} = 0 \]

2. Faraday's Laws of Electromagnetic Induction

Faraday's First Law

Whenever there is a change of magnetic flux linked with a circuit, or whenever a moving conductor cuts magnetic flux, an emf is induced in it.

Faraday's Second Law

The magnitude of induced emf is equal to the rate of change of magnetic flux:

\[ \mathcal{E} = \left| \frac{d\Phi}{dt} \right| \]

2.1 Lenz's Law

The direction of the induced emf is such that it opposes the change in magnetic flux that produces it.

\[ \mathcal{E} = -\frac{d\Phi}{dt} \]

The negative sign indicates the opposition to the change in flux, as per Lenz's law.

2.2 Induced Charge Flow

\[ i = \frac{\mathcal{E}}{R} = -\frac{1}{R}\frac{d\Phi}{dt} \]

Key Points:

  • The induced charge is independent of the manner and time in which the flux changes
  • The induced emf and current depend on the time rate of change of flux

2.3 Motional EMF

When a conductor moves in a magnetic field, an emf is induced across it.

\[ \mathcal{E} = \frac{\Delta\Phi}{\Delta t} = Blv \sin \theta \]

In vector notation:

\[ \mathcal{E} = \int (\vec{v} \times \vec{B}) \cdot d\vec{l} \]

2.4 Rotating Coil in Magnetic Field

\[ \Phi = NBS \cos \omega t \]

Where:

  • N = number of turns
  • S = area of the coil
  • ω = angular velocity of rotation

3. Inductance

3.1 Self-Inductance

Self-inductance (L) is the property of a coil that opposes any change in the current flowing through it.

\[ \Phi \propto I \quad \Rightarrow \quad \Phi = LI \]

Where L is the self-inductance of the coil.

Self-Inductance of a Coil

\[ L = \frac{\Phi}{I} = \frac{\mu_0}{4\pi}(2\pi^2N^2R) = \frac{1}{2}\mu_0\pi N^2R \]

3.2 Energy Stored in an Inductor

\[ U = \frac{1}{2}LI^2 \]

Energy Density of Magnetic Field

\[ u = \frac{B^2}{2\mu_0} \]

3.3 L-R Circuit

Growth of Current
\[ I = I_0 \left( 1 - e^{-t/\tau} \right) \]

Where:

  • I0 = E/R (final current)
  • τ = L/R (time constant)

At t = τ, I = I0(1 - e-1) = 0.63 I0

Decay of Current
\[ I = I_0 e^{-t/\tau} \]

At t = τ, I = I0/e = 0.37 I0

3.4 Mutual Induction

Mutual induction is the phenomenon where an induced emf appears in one circuit due to changes in the magnetic field produced by a nearby circuit.

\[ \Phi_2 = MI_1 \] \[ \mathcal{E}_2 = M \frac{dI_1}{dt} \]

Where M is the coefficient of mutual induction.

\[ M = k\sqrt{L_1 L_2} \]

Where k is the coefficient of coupling (0 ≤ k ≤ 1).

3.5 Grouping of Coils

Coils in Series
\[ L_s = L_1 + L_2 \]
Coils in Parallel
\[ \frac{1}{L_p} = \frac{1}{L_1} + \frac{1}{L_2} \]

4. Alternating Current Circuits

4.1 Alternating Current

\[ I = I_{max} \sin \omega t \]

Where:

  • I = instantaneous current
  • Imax = maximum current
  • ω = angular frequency = 2πf

4.2 RMS and Average Values

RMS Value
\[ I_{rms} = \frac{I_{max}}{\sqrt{2}} = 0.707 I_{max} \] \[ V_{rms} = \frac{V_{max}}{\sqrt{2}} = 0.707 V_{max} \]
Average Value
\[ I_{av} = \frac{2I_{max}}{\pi} = 0.637 I_{max} \]

4.3 AC Circuit Components

Circuit Type Current Expression Impedance/Reactance Phase Relationship Power
Pure Resistive i = Imax sin ωt Z = R Voltage and current in phase P = VrmsIrms
Pure Inductive i = Imax sin(ωt - π/2) XL = ωL Current lags voltage by 90° Zero (wattless)
Pure Capacitive i = Imax sin(ωt + π/2) XC = 1/ωC Current leads voltage by 90° Zero (wattless)
RL Series - Z = √(R² + XL²) Current lags voltage P = VrmsIrms cos φ
RC Series - Z = √(R² + XC²) Current leads voltage P = VrmsIrms cos φ
RLC Series - Z = √[R² + (XL - XC)²] Depends on XL and XC P = VrmsIrms cos φ

4.4 Series Resonance Circuit

At resonance, the inductive and capacitive reactances become equal, resulting in minimum impedance and maximum current.

\[ X_L = X_C \quad \Rightarrow \quad \omega L = \frac{1}{\omega C} \] \[ \omega_r = \frac{1}{\sqrt{LC}} \quad \Rightarrow \quad f_r = \frac{1}{2\pi\sqrt{LC}} \]

Q-Factor (Quality Factor)

\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]

Key Points at Resonance:

  • Impedance is minimum: Z = R
  • Current is maximum
  • Voltage and current are in phase (φ = 0)
  • Power factor is unity (cos φ = 1)

5. Transformers

A transformer is a device that transfers electrical energy from one circuit to another through electromagnetic induction, typically with a change in voltage and current.

5.1 Transformer Principle

\[ \mathcal{E}_1 = -N_1 \frac{d\Phi_B}{dt} \quad \text{and} \quad \mathcal{E}_2 = -N_2 \frac{d\Phi_B}{dt} \]
\[ \frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{N_1}{N_2} \]
\[ \frac{V_2}{V_1} = \frac{N_2}{N_1} \]

Where:

  • V1, V2 = terminal voltages of primary and secondary
  • N1, N2 = number of turns in primary and secondary

5.2 Efficiency of Transformer

\[ \eta = \frac{\text{Output Power}}{\text{Input Power}} \]

Energy Losses in Transformers:

  • Primary resistance (copper loss)
  • Hysteresis in the core
  • Eddy currents in the core
  • Flux leakage

Modern transformers can achieve efficiencies of up to 99%.