JEE Physics Alternating Current Notes Set 04

Alternating Current Basics

An alternating voltage \( V = V_0\sin\omega t \) drives a current that changes direction periodically, with \( \omega = 2\pi f \). The rms value is the DC value that gives the same heating: \( I_{rms} = I_0/\sqrt2 \) and \( V_{rms} = V_0/\sqrt2 \). For a sum \( a\sin\omega t + b\cos\omega t \) the amplitude is \( \sqrt{a^2+b^2} \). Meters for AC read rms values.

Pure R, L and C Circuits

1. Resistor: current in phase with voltage, \( I = V/R \).
2. Inductor: reactance \( X_L = \omega L \), current lags voltage by 90°, and the average power is zero.
3. Capacitor: reactance \( X_C = 1/\omega C \), current leads voltage by 90°, and the average power is zero.
4. The instantaneous power in a pure L or C oscillates at twice the supply frequency, so it is zero on average.

Series LCR Circuit

1. Impedance: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). Voltages: \( V = \sqrt{V_R^2 + (V_L - V_C)^2} \).
2. Phase angle: \( \tan\phi = (X_L - X_C)/R \). If \( X_L > X_C \) the current lags, otherwise it leads.
3. Resonance occurs at \( \omega_0 = 1/\sqrt{LC} \). Z is minimum (equal to R), current is maximum, and \( V_L = V_C \) cancel.
4. Quality factor \( Q = \dfrac{\omega_0L}{R} = \dfrac1R\sqrt{\dfrac LC} \). A larger Q means a sharper resonance and a narrower band width.
5. Time constant of an LR circuit \( \tau = L/R \); with AC the power factor can be written \( R/\sqrt{R^2+\omega^2L^2} = 1/\sqrt{1+(\omega\tau)^2} \).

Illustration 1. A 160 V, 60 W lamp is run from a 200 V ac mains using a capacitor in series. Find the voltage across the capacitor.
Answer:
The lamp voltage and the capacitor voltage are 90° out of phase.
\( V_C = \sqrt{200^2 - 160^2} = 120 \) V.

Power in AC Circuits

1. Average power: \( P = V_{rms}I_{rms}\cos\phi \), where \( \cos\phi = R/Z \) is the power factor.
2. Wattless current is \( I_{rms}\sin\phi \) and its power is zero.
3. Power in an LC circuit, a pure L or a pure C is zero. At resonance the power factor is 1.

LC Oscillations

1. A capacitor charged to \( q_0 \) discharges through an inductor. Charge and current oscillate with \( \omega = 1/\sqrt{LC} \) and period \( T = 2\pi\sqrt{LC} \).
2. Energy shuttles between the electric field (\( q^2/2C \)) and the magnetic field (\( \tfrac12Li^2 \)). The total is constant.
3. The analogy with a spring-block system: \( m\leftrightarrow L \), \( k\leftrightarrow 1/C \), \( x\leftrightarrow q \), \( v\leftrightarrow i \).

Electromagnetic Waves

1. Accelerating charges produce electromagnetic waves. The electric and magnetic fields are perpendicular to each other and to the direction of propagation, and oscillate in phase: the direction of travel is along \( \vec E\times\vec B \).
2. Speed in vacuum \( c = 1/\sqrt{\mu_0\epsilon_0} = 3\times10^8 \) m/s, and \( E_0/B_0 = c \). In a medium \( v = c/n \).
3. Energy is shared equally between the E and B fields. Intensity \( I = \tfrac12\epsilon_0E_0^2c \). Radiation pressure is \( I/c \) on a perfect absorber and \( 2I/c \) on a perfect reflector.
4. Displacement current \( i_d = \epsilon_0\,d\phi_E/dt \) carries the current across a capacitor gap, and equals the conduction current in the wires.
5. Spectrum in order of increasing frequency: radio waves, microwaves, infrared, visible, ultraviolet, X-rays and gamma rays. The ozone layer absorbs most of the harmful ultraviolet, and the greenhouse effect is due to infrared radiation trapped by the atmosphere.