Electromagnetic Induction and Alternating Current

Magnetic Flux: $\Phi_B = \int \vec{B}\cdot d\vec{A} = BA\cos\theta$ for uniform field. SI unit: weber (Wb) = T·m².

Faraday's Law: Induced EMF equals negative rate of change of flux: $$\varepsilon = -\frac{d\Phi_B}{dt}$$ For $N$ turns: $\varepsilon = -N\,d\Phi_B/dt$.

Lenz's Law: Induced current opposes the change in flux (consequence of energy conservation). Sign of negative in Faraday's law.

Motional EMF: A conductor of length $L$ moving with velocity $v$ perpendicular to $\vec{B}$: $$\varepsilon = BLv$$ General form: $\varepsilon = \int(\vec{v}\times\vec{B})\cdot d\vec{l}$.

Induced electric field (non-conservative): $\oint \vec{E}\cdot d\vec{l} = -d\Phi_B/dt$.

Eddy Currents: Currents induced in bulk conductors by changing flux; dissipate as heat (used in induction heating, electromagnetic braking; reduced by lamination in transformer cores).

Rotating Coil in Magnetic Field: $N$ turns, area $A$, angular velocity $\omega$ in field $B$: $$\varepsilon = NAB\omega\sin\omega t, \quad \varepsilon_0 = NAB\omega$$ Basis of AC generator.

Self-Inductance ($L$): EMF induced in a coil due to change in its own current. $$\varepsilon = -L\frac{dI}{dt}, \quad L = \frac{N\Phi}{I}$$ SI unit: henry (H) = Wb/A = V·s/A.

Self-Inductance of Common Configurations:

Mutual Inductance ($M$): EMF induced in one coil due to current change in another. $$\varepsilon_2 = -M\frac{dI_1}{dt}, \quad M = \frac{N_2\Phi_{21}}{I_1}$$

For two coils with self-inductances $L_1, L_2$ and coupling coefficient $k$: $$M = k\sqrt{L_1 L_2}, \quad 0 \leq k \leq 1$$

Series/Parallel Combination of Inductors (no mutual coupling):

• Series: $L_{\text{eq}} = L_1 + L_2$
• Parallel: $1/L_{\text{eq}} = 1/L_1 + 1/L_2$

With mutual: $L_{\text{series}} = L_1 + L_2 \pm 2M$.

Energy Stored in Inductor: $$U = \frac{1}{2}LI^2$$

Magnetic Energy Density: $u_B = B^2/(2\mu_0)$.

LR Circuit (growth): $I = (\varepsilon/R)(1 - e^{-Rt/L})$. Time constant: $\tau = L/R$.

LR Circuit (decay): $I = I_0 e^{-Rt/L}$.

AC Voltage/Current: $V = V_0\sin\omega t$, $I = I_0\sin(\omega t \pm \phi)$. Frequency $f = \omega/(2\pi)$; period $T = 1/f$.

Mean and RMS Values: For sinusoidal AC over one full cycle:

• Average value over full cycle = $0$
• Average over half-cycle: $\bar V = 2V_0/\pi \approx 0.637 V_0$
• RMS value: $V_{\text{rms}} = V_0/\sqrt{2} \approx 0.707 V_0$

RMS value is the DC equivalent that produces same heating. Power: $P = V_{\text{rms}}I_{\text{rms}}\cos\phi$.

AC Through Single Elements:

Reactance $X_L, X_C$ measured in ohms. $X_L$ increases with frequency; $X_C$ decreases.

Inductor blocks high $f$; capacitor blocks low $f$ (DC).

Power in Pure Reactive Element: $\langle P\rangle = 0$ (over full cycle). Pure $L$ or $C$ are "wattless".

Power Factor: $\cos\phi = R/Z$, where $Z$ = impedance. $\phi$ = phase angle.

Series LCR Circuit with AC source: $$Z = \sqrt{R^2 + (X_L - X_C)^2}$$ $$\tan\phi = \frac{X_L - X_C}{R}$$

Phase angle $\phi$: positive if inductive ($X_L > X_C$), negative if capacitive.

Current: $I_{\text{rms}} = V_{\text{rms}}/Z$. Voltage drops: $V_R = IR$, $V_L = IX_L$, $V_C = IX_C$ (vector addition needed since not in phase).

Resonance occurs when $X_L = X_C$: $$\omega_0 = \frac{1}{\sqrt{LC}}, \quad f_0 = \frac{1}{2\pi\sqrt{LC}}$$

At resonance: $Z = R$ (minimum), $I$ = maximum, $\phi = 0$ (in phase), power factor = 1.

Quality Factor (Q): $$Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$$ High $Q$ → sharp resonance peak (narrow bandwidth). $Q$ = $\omega_0/\Delta\omega$ = $f_0$/bandwidth.

Average Power: $$P = V_{\text{rms}}I_{\text{rms}}\cos\phi = I_{\text{rms}}^2 R$$

At resonance, power is maximum ($\cos\phi = 1$).

Transformer: Two coils (primary, secondary) on common iron core. For ideal transformer: $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

Step-up: $N_s > N_p$ → voltage increases, current decreases. Step-down: opposite.

Power conserved (ideal): $V_p I_p = V_s I_s$. Real transformer efficiency $\eta = P_{\text{out}}/P_{\text{in}}$ (typically $95-99\%$).

Losses in transformer: copper loss ($I^2R$), iron loss (eddy + hysteresis), flux leakage, humming.