To an ac power supply of 220 V at 50 Hz , a resistor of , a capacitor of reactance and an inductor of reactance are connected in series. The corresponding current in the circuit and the phase angle between the current and the voltage is, respectively
- A15.6 A and
- B15.6 A and
- C7.8 A and
- D7.8 A and
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Correct answer: D
To determine the current in the circuit and the phase angle between the current and voltage, follow these steps:
Components and their Reactance:
Inductive reactance, $ X_L = 45 \, \Omega $
Capacitive reactance, $ X_C = 25 \, \Omega $
Resistance, $ R = 20 \, \Omega $
Calculate Current (I):
The current through the series circuit can be calculated using the formula:
$ I = \frac{V}{\sqrt{(X_L - X_C)^2 + R^2}} $
Substituting the given values:
$ I = \frac{220}{\sqrt{(45 - 25)^2 + 20^2}} = \frac{220}{\sqrt{20^2 + 20^2}} $
Simplifying further:
$ I = \frac{220}{\sqrt{400 + 400}} = \frac{220}{\sqrt{800}} = \frac{220}{2\sqrt{200}} = \frac{220}{2\sqrt{100 \times 2}} = \frac{220}{20\sqrt{2}} = \frac{11}{\sqrt{2}} $
Therefore, the current is approximately:
$ I = 7.779 \, \text{A} \, (\approx 7.8 \, \text{A}) $
Calculate Phase Angle ($\phi$):
The phase angle can be calculated using the tangent of the angle:
$ \tan \phi = \frac{X_L - X_C}{R} = \frac{45 - 25}{20} = \frac{20}{20} = 1 $
Hence, the phase angle $\phi$ is:
$ \phi = 45^{\circ} $
Thus, the current in the circuit is approximately 7.8 A, and the phase angle is $45^{\circ}$.
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