Electromagnetic Induction and Alternating Currents revision notes

Electromagnetic Induction (EMI)

Alternating Current (AC)

Motional EMF in Straight and Rotating Conductors

When a straight rod of length $l$ translates with velocity $v$ perpendicular to a uniform magnetic field $B$, free electrons experience a magnetic Lorentz force, creating a potential difference $\varepsilon = Blv$. When a rod rotates about one end in a perpendicular magnetic field with angular velocity $\omega$, the induced EMF is $\varepsilon = \frac{1}{2} B \omega R^2$. JEE Tip If a metal wheel with $N$ metallic spokes rotates in a uniform magnetic field, the EMF between the axle and the rim remains exactly $\frac{1}{2} B \omega R^2$, regardless of the number of spokes. All spokes act as identical voltage sources connected in parallel!

Self and Mutual Inductance

For a long solenoid, self-inductance heavily depends on its geometry (turns, area, length) and the core medium's permeability ($\mu_r$). Mutual inductance between two coaxial coils is symmetric ($M_{12} = M_{21}$). JEE Tip Mutual inductance strongly depends on the separation distance and the relative orientation (angle) between the two coils, not just their sizes and turns.

Magnetic Energy

Work must be done against the back-EMF to establish a current in an inductor. This work is stored as magnetic potential energy $U_B = \frac{1}{2}LI^2$ (analogous to mechanical kinetic energy $K = \frac{1}{2}mv^2$). The magnetic energy density is $u_B = \frac{B^2}{2\mu_0}$.

Transient LR Circuits (JEE Advanced Add-on)

• Growth: Current grows as $I(t) = \frac{V}{R} (1 - e^{-Rt/L})$.
• Decay: Current decays as $I(t) = I_0 e^{-Rt/L}$. JEE Tip At $t=0$ (just after closing the switch), an unenergized inductor offers infinite resistance (acts as an open wire). At $t \to \infty$ (steady state), an inductor offers zero resistance to DC (acts as a simple connecting wire).

Induced Electric Field (JEE Advanced Add-on)

A time-varying magnetic field produces an induced electric field. This explains how an EMF is generated in a stationary loop. These induced fields are non-conservative (they form closed loops), meaning work done moving a charge in a closed loop is non-zero.

AC Voltage and Current Applied to Circuit Elements

• Resistor: Current and voltage are perfectly in phase ($\phi = 0$). Power factor $\cos \phi = 1$.
• Inductor: Current lags the voltage by exactly $\pi/2$. Power factor $\cos \phi = 0$.
• Capacitor: Current leads the voltage by exactly $\pi/2$. Power factor $\cos \phi = 0$.

Series LCR Circuit & Phasor Analysis

The instantaneous voltages algebraically add up: $v = v_R + v_L + v_C$. However, their RMS or peak values must be added using vector addition: $V = \sqrt{V_R^2 + (V_L - V_C)^2}$. JEE Tip In an AC circuit, if you measure $150\text{V}$ across a resistor and $160\text{V}$ across a capacitor, the source voltage IS NOT $310\text{V}$. The voltages are $90^\circ$ out of phase, so the source is $\sqrt{150^2 + 160^2}$.

Resonance in LCR Circuits

Resonance cannot occur in simple RL or RC circuits; it explicitly requires both energy-storing elements (L and C) so their out-of-phase voltages can cancel. JEE Tip At resonance, the voltages across the inductor and capacitor can be significantly larger than the source voltage. This is called voltage magnification, and the magnification factor is the Quality Factor ($Q$).

Power Factor Improvement

In long-distance power transmission, a low power factor implies large current is needed to deliver the same power ($P = VI \cos \phi$), causing massive $I^2R$ transmission line losses. JEE Tip Power factor is improved (brought closer to 1) by connecting a capacitor of appropriate value in parallel to the circuit. This provides a leading wattless current ($I'_q$) that completely neutralizes the lagging inductive wattless current ($I_q$).

Transformers

Operates purely on mutual induction to step-up or step-down AC voltage.

• Step-up: $N_s > N_p \Rightarrow V_s > V_p \Rightarrow I_s < I_p$. (Stepping up voltage steps down current, conserving energy).

• Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA \cos \theta$ (Unit: Weber (Wb)).
• Faraday's Law: $\varepsilon = -N \frac{d\Phi_B}{dt}$ (Unit: Volt (V)).
• Motional EMF (Translating): $\varepsilon = B l v$.
• Motional EMF (Rotating): $\varepsilon = \frac{1}{2} B \omega R^2$.
• Self-Induced Back EMF: $\varepsilon = -L \frac{dI}{dt}$ (Unit of L: Henry (H)).
• Self-Inductance of a Solenoid: $L = \mu_r \mu_0 n^2 A l$ (where $n = N/l$).
• Mutual Inductance of Two Solenoids: $M_{12} = M_{21} = \mu_r \mu_0 n_1 n_2 \pi r_1^2 l$ (where $r_1$ is inner radius).
• Magnetic Energy Stored: $U_B = \frac{1}{2} L I^2$.
• Magnetic Energy Density: $u_B = \frac{B^2}{2\mu_0}$.
• AC Generator EMF: $\varepsilon = N B A \omega \sin(\omega t) = \varepsilon_0 \sin(\omega t)$.
• RMS Values: $I_{rms} = \frac{I_m}{\sqrt{2}}$, and $V_{rms} = \frac{V_m}{\sqrt{2}}$.
• Reactance: $X_L = \omega L$ and $X_C = \frac{1}{\omega C}$ (Unit: Ohm ($\Omega$)).
• Impedance in Series LCR: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ (Unit: Ohm ($\Omega$)).
• Phase Angle in Series LCR: $\tan \phi = \frac{X_L - X_C}{R}$.
• Resonant Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$.
• Quality Factor: $Q = \frac{1}{R} \sqrt{\frac{L}{C}}$.
• Average Power in AC: $P_{avg} = V_{rms} I_{rms} \cos \phi = I_{rms}^2 R$.
• Transformer Ratio: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$.

• Motional EMF Formula ($Blv$): Valid only if $\vec{B}$, $\vec{v}$, and $\vec{l}$ are mutually perpendicular, and the field is uniform.
• Mutual Inductance ($M = \mu_0 n_1 n_2 \pi r_1^2 l$): Assumes the inner solenoid is much longer/smaller so it entirely resides in the uniform region of the outer solenoid, effectively neglecting edge effects.
• Transformer Equations: Valid strictly under ideal assumptions: 100% efficiency, negligible primary resistance, identical flux linking both coils (zero leakage), and small secondary currents.
• Transformers and DC: Transformers absolutely DO NOT work on DC. Mutual induction strictly requires a varying magnetic flux ($\frac{d\Phi}{dt} \neq 0$).

• Open Circuit EMF: If an open loop is placed in a changing magnetic field, an EMF is still induced across the ends, even though no current flows!
• Lenz's Law & Energy Conservation: A common error is assuming induced current flows in the direction of the external magnetic field. It actually flows in the direction that creates its own magnetic field to oppose the change in external flux. Reversing this rule would result in a perpetual motion machine, violating conservation of energy.
• Sign of Inductive EMF: In $\varepsilon = -L \frac{dI}{dt}$, if current is decreasing ($dI/dt$ is negative), $\varepsilon$ becomes positive, meaning the inductor acts as a battery supporting the current flow.
• Trap in Flux Change Calculation: When a coil is flipped $180^\circ$ in a magnetic field, the change in flux is $\Delta \Phi = (-BA) - (BA) = -2BA$, not zero.
• "What if" Variable Area: If a loop is pulled out of a magnetic field, the induced EMF is constant only for a rectangular loop moving at constant velocity. For a circular loop, the rate of change of area ($dA/dt$) is non-constant, so the induced EMF varies over time.

• Voltage/Current vs Time in Pure Resistor: Both $v$ and $i$ sine waves peak and cross zero simultaneously ($\phi=0$).
• Voltage/Current vs Time in Pure Inductor/Capacitor: The current sine wave is shifted horizontally by $T/4$ (or $\pi/2$ radians) relative to the voltage wave.
• Impedance Triangle: A right-angled triangle with base $R$, perpendicular $(X_L - X_C)$, and hypotenuse $Z$. The angle between $R$ and $Z$ is $\phi$. This instantly visualizes power factor ($\cos \phi = R/Z$).
• Resonance Curve ($I_m$ vs $\omega$): A bell-like curve peaking at $\omega_0$. For lower $R$, the peak is taller and sharper (higher Q-factor). For higher $R$, the peak is shorter and broader.

1. AC Generator EMF

1. A coil of $N$ turns and area $A$ rotates with angular velocity $\omega$ in field $B$.
2. The angle between area vector $\vec{A}$ and $\vec{B}$ at time $t$ is $\theta = \omega t$.
3. The instantaneous flux is $\Phi_B = NBA \cos(\omega t)$.
4. By Faraday's Law, $\varepsilon = -d\Phi_B / dt = -(-NBA\omega \sin(\omega t))$.
5. Thus, $\varepsilon = \varepsilon_0 \sin(\omega t)$, where maximum EMF $\varepsilon_0 = NBA\omega$.

2. Deriving Motional EMF via Lorentz Force

1. A rod of length $l$ moves at velocity $v$ in perpendicular field $B$.
2. Free electrons move with velocity $v$, experiencing magnetic Lorentz force $F_m = evB$ directed along the rod.
3. This pushes electrons to one end, leaving the other positive, creating an internal electrostatic field $E$.
4. At steady state, electric force equals magnetic force: $eE = evB \implies E = vB$.
5. The potential difference is $\varepsilon = E \cdot l = B l v$. JEE Tip This shows motional EMF is fundamentally a redistribution of charges leading to a measurable potential difference.

3. Phasor Method for LCR Circuits

1. Assume the steady-state current is $i = i_m \sin(\omega t + \phi)$.
2. Draw voltage phasors: $V_R$ parallel to $I$, $V_L$ $90^\circ$ ahead of $I$, $V_C$ $90^\circ$ behind $I$.
3. Combine $V_L$ and $V_C$ into a single vector $(V_L - V_C)$.
4. Use Pythagoras with $V_R$ and $(V_L - V_C)$ to find source voltage $V_m = \sqrt{V_{Rm}^2 + (V_{Lm} - V_{Cm})^2}$.
5. Divide by $I_m$ to deduce $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

• Motional EMF in Arbitrary Shapes: The EMF induced across a curved wire translating in a uniform magnetic field depends only on the straight-line displacement vector between its endpoints.
• Time-Varying Magnetic Fields: Finding the induced non-conservative electric field inside and outside a cylindrical region of changing magnetic field ($E_c \cdot 2\pi r = A_{enclosed} \cdot \frac{dB}{dt}$).
• LCR Resonance and Q-Factor: Calculating bandwidth, half-power frequencies, and analyzing how changing $C$ or $L$ affects the glowing brightness of a bulb in series.
• Area Under e-t Graph: The integral of induced EMF with respect to time gives the total change in flux ($\int \varepsilon dt = \Delta \Phi$), and total charge flown is $q = \frac{\Delta \Phi}{R}$, which is astonishingly independent of the time taken or speed of the flux change.