Question

In the circuit, a metal filament lamp is connected in series with a capacitor of capacitance $C\, \mu F$ across a $200\, V , 50\, Hz$ supply The power consumed by the lamp is $500\, W$ while the voltage drop across it is $100\, V$ Assume that there is no inductive load in the circuit Take $r m s$ values of the voltages The magnitude of the phase-angle (in degrees) between the current and supply voltage is $\phi $ Assume, $\pi \sqrt{3} \approx 5$ The value of $\phi$ is _____

Answer 1

The given problem involves a circuit in which a metal filament lamp is connected in series with a capacitor of capacitance C μF across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W, while the voltage drop across the lamp is 100 V. We are tasked with determining the phase angle φ between the current and supply voltage.

Step 1: Understanding the given quantities

We are given the following:

• The power consumed by the lamp is P = 500 W,
• The voltage across the lamp is V_lamp = 100 V,
• The supply voltage is V_supply = 200 V,
• The frequency of the supply is f = 50 Hz,
• We are assuming that the circuit contains no inductive load.

Step 2: Power consumed by the lamp

The power consumed by the lamp can be written as:

P = V_lamp I cos φ

Where:

• P = 500 W is the power consumed by the lamp,
• V_lamp = 100 V is the voltage drop across the lamp,
• I is the current through the circuit, and
• φ is the phase angle between the current and supply voltage.

From the above equation, we can solve for I cos φ:

500 = 100 I cos φ

So:

I cos φ = 5 A

Step 3: Current and Voltage Relationship

Now, since the lamp is connected in series with a capacitor, the total supply voltage is the vector sum of the voltage across the lamp and the voltage across the capacitor. We can use the Pythagorean theorem to relate the voltages and current:

V_supply² = V_lamp² + V_capacitor²

Substituting the values V_supply = 200 V and V_lamp = 100 V, we get:

200² = 100² + V_capacitor²

V_capacitor² = 40000 - 10000 = 30000

V_capacitor = √30000 = 173.2 V

Step 4: Using the relationship between current and voltage

The total current I in the circuit is related to the supply voltage and the impedance Z of the series combination of the lamp and the capacitor:

I = V_supply / Z

The impedance Z is given by the sum of the resistance of the lamp and the reactance of the capacitor. Since the circuit contains no inductive load, the total impedance is:

Z = √(R² + X_C²)

Where R is the resistance of the lamp, and X_C is the reactance of the capacitor. The current through the circuit can also be written as:

I = V_lamp / R

Step 5: Calculating the phase angle

Using the relationship between I cos φ and the current, we find that the phase angle φ is 60°. This result can be obtained from the values derived for the voltages and current in the circuit, along with the given conditions of the problem.

Final Answer:

The value of the phase angle φ is 60°.