JEE Physics Alternating Current Notes Set 02

Alternating Voltage and Current

An alternating emf varies with time as a sine wave: \( e = E_0\sin\omega t \), where \( \omega = 2\pi f = \frac{2\pi}{T} \). The peak value is \( E_0 \) and the frequency f is the number of cycles per second.

1. Mean (average) value over a full cycle is zero. Over a half cycle it is \( \frac{2}{\pi}I_0 = 0.637I_0 \).
2. Root mean square (rms) value: \( I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707I_0 \), and similarly \( V_{rms} = \frac{V_0}{\sqrt{2}} \). AC meters and household ratings (such as 220 V) are rms values, so the peak of a 220 V supply is about 311 V.
3. The rms value is the dc value that would produce the same heating in a resistor. For a sine wave, the current reaches its rms value one-eighth of a cycle after zero (at 45 degrees).

AC Through a Single Element

For an inductor the voltage across it is \( L\frac{dI}{dt} \), so a current \( I_0\sin\omega t \) gives \( V = \omega LI_0\cos\omega t \). Reactance is measured in ohms but, unlike resistance, it does not dissipate energy. An inductor passes low frequencies and blocks high ones, while a capacitor blocks dc and passes high frequencies.

Series LCR Circuit

1. Voltages across L and C are in opposite phase, so net reactance is \( X = X_L - X_C \). Voltage across R is in phase with the current.
2. Impedance \( Z = \sqrt{R^2 + (X_L - X_C)^2} \) and source voltage \( V = \sqrt{V_R^2 + (V_L - V_C)^2} \).
3. Phase angle \( \tan\phi = \frac{X_L - X_C}{R} \). If \( X_L > X_C \) the circuit is inductive and the current lags the voltage. If \( X_C > X_L \) it is capacitive and the current leads.
4. The voltage across an individual L or C can be larger than the source voltage. In an R-L circuit \( V^2 = V_R^2 + V_L^2 \), for example 12 V and 16 V give 20 V.

Resonance

1. At resonance \( X_L = X_C \), so \( \omega_0 = \frac{1}{\sqrt{LC}} \) and \( f_0 = \frac{1}{2\pi\sqrt{LC}} \). Resistance does not change the resonant frequency.
2. Impedance is minimum (\( Z = R \)), current is maximum \( \frac{V}{R} \), phase angle is zero and power factor is 1. Average power equals apparent power and the wattless current is zero.
3. Voltages across L and C are equal and opposite, each equal to \( I X_L \), which may far exceed the source voltage.
4. Quality factor \( Q = \frac{\omega_0L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \). A high Q means a sharp resonance curve, used in radio tuning.
5. Changing C to one-fourth requires L to become four times for the same resonant frequency, because \( LC \) must stay constant.

Power in an AC Circuit

1. Average power \( P = V_{rms}I_{rms}\cos\phi \), where \( \cos\phi = \frac{R}{Z} \) is the power factor.
2. Equivalent form: \( P = \frac{V_{rms}^2R}{R^2 + (X_L - X_C)^2} \).
3. Wattless current is the component \( I_{rms}\sin\phi \), which does no work. A choke coil with high L and low R reduces ac current with very little energy loss.
4. Power is maximum at resonance, equal to \( \frac{V_{rms}^2}{R} \).

LC Oscillations

A charged capacitor connected to an inductor oscillates at \( \omega = \frac{1}{\sqrt{LC}} \). Energy keeps swapping between the electric field of the capacitor and the magnetic field of the inductor. Total energy \( \frac{Q^2}{2C} = \frac{1}{2}LI_0^2 \) stays constant. The energy is shared equally when the charge is \( \frac{Q}{\sqrt{2}} \).

Worked Examples

1. A coil with \( X_L = 31\ \Omega \) and \( R = 8\ \Omega \) is in series with a capacitor of \( X_C = 25\ \Omega \). Net reactance is 6 ohm, so \( Z = \sqrt{64 + 36} = 10\ \Omega \) and power factor is \( \frac{8}{10} = 0.8 \).
2. An LCR circuit has \( V_R = 8 \) V, \( V_L = 6 \) V and \( V_C = 12 \) V. Source voltage \( = \sqrt{8^2 + 6^2} = 10 \) V. The circuit is capacitive, so the current leads the voltage, and the power factor is 0.8.
3. A series circuit with \( L = 27 \) mH and \( C = 30\ \mu F \) has \( \omega_0 = \frac{1}{\sqrt{8.1\times10^{-7}}} \approx 1.1\times10^3 \) rad/s.
4. A 30 \( \mu \)F capacitor on a 150 V, 60 Hz supply has \( X_C \approx 88\ \Omega \) and carries about 1.7 A rms.

Common Mistakes

1. Using peak values in the power formula instead of rms values.
2. Confusing angular frequency \( \omega \) (rad/s) with frequency f (Hz).
3. Adding \( V_R \), \( V_L \) and \( V_C \) arithmetically instead of as phasors.
4. Forgetting that a galvanometer for dc shows zero for ac because the average of a sine wave is zero.