The angular frequency of alternating current in a L-C-R circuit is 100 rad s$$^{-1}$$. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser.

Solution & Explanation

1. Find the Circuit Current ($$I$$)

From the given figure, we can observe the values for the top horizontal branch:

• Resistance of the resistor, $$R = 60 \,\, \Omega$$
• Potential drop across this resistor, $$V_R = 15 \,\, \text{V}$$

Since the alternating components share the same series current network loop, we can apply Ohm's Law across this specific resistor to find the RMS current ($$I$$) flowing through the entire circuit:

$$I = \frac{V_R}{R} = \frac{15 \,\, \text{V}}{60 \,\, \Omega} = 0.25 \,\, \text{A}$$

2. Calculate the Inductance ($$L$$) of the Coil

From the diagram, the potential drop across the inductor is $$V_L = 20 \,\, \text{V}$$. The inductive reactance ($$X_L$$) can be expressed as:

$$V_L = I \cdot X_L \implies X_L = \frac{V_L}{I}$$

$$X_L = \frac{20 \,\, \text{V}}{0.25 \,\, \text{A}} = 80 \,\, \Omega$$

We know that inductive reactance is related to the angular frequency ($$\omega = 100 \,\, \text{rad s}^{-1}$$) and inductance ($$L$$) by the formula $$X_L = \omega L$$:

L = $$\frac{X_L}{\omega}$$ = $$\frac{80}{100} = 0.8 \,\, \text{H}$$

3. Calculate the Capacitance ($$C$$) of the Condenser

From the diagram, the potential drop across the capacitor is $$V_C = 10 \,\, \text{V}$$. The capacitive reactance ($$X_C$$) is given by:

$$V_C = I \cdot X_C \implies X_C = \frac{V_C}{I}$$

$$X_C = \frac{10 \,\, \text{V}}{0.25 \,\, \text{A}} = 40 \,\, \Omega$$

The relationship between capacitive reactance, angular frequency ($$\omega$$), and capacitance ($$C$$) is defined by $$X_C = \frac{1}{\omega C}$$:

$$C = \frac{1}{\omega \cdot X_C} = \frac{1}{100 \times 40} = \frac{1}{4000} \,\, \text{F}$$

To convert Farads into microfarads ($$\mu\text{F}$$), multiply the result by $$10^6$$:

$$C = \frac{10^6}{4000} \,\, \mu\text{F} = \frac{1000}{4} \,\, \mu\text{F} = 250 \,\, \mu\text{F}$$

Correct Option: D ($$0.8 \,\, \text{H and } 250 \,\, \mu\text{F}$$)