The given circuit has an AC voltage source with voltage \( V = 400 \sin(10 \pi t) \) V, an inductor \( L \), a capacitor \( C \), and a resistor \( R = 200 \, \Omega \) in series. The inductance \( L = 80 \, \text{V} \), and the capacitance \( C = 80 \, \text{V} \).
Step 1: Voltage across the components.
The voltage across the resistor \( R \) can be calculated using Ohm's law:
\[
V_R = I R
\]
where \( I \) is the current in the circuit.
The total impedance \( Z \) of the series circuit is given by:
\[
Z = \sqrt{R^2 + (X_L - X_^2}
\]
where:
- \( X_L = \omega L \) is the inductive reactance,
- \( X_C = \frac{1}{\omega C} \) is the capacitive reactance,
- \( \omega = 10 \pi \) is the angular frequency.
Substituting the values:
\[
X_L = \omega L = 10 \pi \times 80 = 800 \pi \, \Omega, \quad X_C = \frac{1}{\omega C} = \frac{1}{10 \pi \times 80} = \frac{1}{800 \pi} \, \Omega
\]
Now, calculate the total impedance:
\[
Z = \sqrt{R^2 + (X_L - X_^2} = \sqrt{200^2 + (800 \pi - \frac{1}{800 \pi})^2}
\]
Step 2: Current in the circuit.
The current \( I \) can be calculated from the voltage and impedance:
\[
I = \frac{V_{\text{max}}}{Z} = \frac{400}{Z}
\]
Step 3: Voltage across the resistor.
Once the current \( I \) is calculated, we can use Ohm's law to find the voltage across the resistor:
\[
V_R = I R
\]
Step 4: Phase difference.
The phase difference \( \Delta \phi \) between the inductor and the capacitor is given by:
\[
\Delta \phi = \tan^{-1}\left( \frac{X_L - X_C}{R} \right)
\]
Step 5: Conclusion.
Using the above formulas and calculations, we can obtain the current, voltage across the resistor, and the phase difference between the inductor and the capacitor.
