Question

In the given circuit, find (i) voltage across resistance, (ii) current in the circuit, (iii) phase difference between voltages across inductance and capacitance.

Hint:

When $X_L = X_C$, the circuit is at resonance: net reactance = 0, current is maximum.

Answer 1

Step 1: Identify circuit elements.
$R = 400 \, \Omega$, $V = 200\cos(50\pi t)$ (rms value $= 200/\sqrt{2}$), inductive reactance $X_L = 180 \, \Omega$, capacitive reactance $X_C = 180 \, \Omega$.
Step 2: Net reactance.
Since $X_L = X_C$, they cancel each other. \[ X = X_L - X_C = 0 \]
Step 3: Impedance.
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{400^2 + 0} = 400 \, \Omega \]
Step 4: Current in circuit.
RMS source voltage $V_{rms} = 200/\sqrt{2} \approx 141.4 \, V$. \[ I = \frac{V_{rms}}{Z} = \frac{141.4}{400} \approx 0.354 \, A \]
Step 5: Voltage across resistance.
\[ V_R = IR = 0.354 \times 400 \approx 141.4 \, V \]
Step 6: Phase difference between $V_L$ and $V_C$.
Since $X_L = X_C$, the voltages are equal but $180^\circ$ out of phase.
Step 7: Conclusion.
- Voltage across $R$: $141.4 \, V$
- Current: $0.354 \, A$
- Phase difference: $180^\circ$