Question

A device Y is connected across an AC source of emf e = e₀ sin ωt. The current through Y is given as i = i₀ sin (ωt + π/2).

1. Identify the device Y and write the expression for its reactance.
2. Draw graphs showing a variation of emf and current with time over one cycle of AC for Y.
3. How does the reactance of the device Y vary with the frequency of the AC? Show graphically.
4. Draw the phasor diagram for device Y.

Answer 1

1. The device Y is a capacitor. Its reactance is `"X"_"C" = 1/(omega"C")`, where ω is the angular frequency of the applied emf and C is the capacitance of the capacitor.

2.

3. `"X"_"C" = 1/(omega"C") = 1/(2pi"fC")`. Thus `"X"_"C" prop 1/"f"`, where f is the frequency of AC.
Suppose C = `(1000/(2pi))`pF

For f = 100 Hz, X_C = 1 × 10⁷ Ω = 10 M Ω;

for f = 200 Hz, X_C = 5 M Ω;

for f = 300 Hz, X_C = `10/3` M Ω;

for f = 400 Hz, X_C = 2.5 M Ω

for f = 500 Hz, X_C = 2 M Ω and so on. 
(1 M Ω = 10⁶ Ω)

           Variation of `1/(omega"C")` with f

4. Phasor diagram for a purely capacitive circuit

The phasor representing the peak emf (e₀) makes an angle wt in an anticlockwise direction with respect to the horizontal axis. As the current leads the voltage by 90°, the phasor representing the peak current (i₀) is turned 90° anticlockwise with respect to the phasor representing emf e₀. The projections of these phasors on the vertical axis give instantaneous values of e and i.