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Step-by-Step Solution
Step 1: Identify the given quantities
• Resistance, $R = 100\,\Omega$
• Inductance, $L = 0.5\,\text{mH} = 0.5 \times 10^{-3}\,\text{H}$
• Capacitance, $C = 0.1\,\text{pF} = 0.1 \times 10^{-12}\,\text{F}$
• AC supply frequency, $f = 50\,\text{Hz}$
• RMS supply voltage, $V = 220\,\text{V}$
Step 2: Calculate the angular frequency
The angular frequency $ \omega $ is given by
$$ \omega = 2\pi f. $$
For $f = 50\,\text{Hz}$,
$$ \omega = 2\pi \times 50 = 100\pi\,\text{rad\,s}^{-1}. $$
Step 3: Calculate the inductive reactance $X_L$
Inductive reactance is given by
$$ X_L = \omega L. $$
Substituting the values:
$$ X_L = 100\pi \times 0.5 \times 10^{-3} = 50\pi \times 10^{-3} = 0.05\pi\,\Omega. $$
Numerically, $0.05\pi \approx 0.157\,\Omega$, which is quite small.
Step 4: Calculate the capacitive reactance $X_C$
Capacitive reactance is given by
$$ X_C = \frac{1}{\omega C}. $$
Substituting the values:
$$ X_C = \frac{1}{100\pi \times 0.1 \times 10^{-12}}
= \frac{1}{100\pi \times 10^{-13}}
= \frac{10^{13}}{100\pi}
= \frac{10^{11}}{\pi}\,\Omega. $$
Numerically, this is on the order of $ \sim 3.18 \times 10^{10}\,\Omega, $ which is extremely large.
Step 5: Compare the magnitudes of $X_L$ and $X_C$
We see that
$$ X_C \gg X_L. $$
Hence, the circuit behaves as a predominantly capacitive circuit because the net reactance is
$$ X = X_L - X_C \approx -X_C \quad (\text{since } X_C \gg X_L), $$
and a negative reactance indicates a capacitive nature.
Step 6: Determine the phase angle
In a series circuit with a dominant capacitive reactance, the current leads the voltage and the net phase angle is approximately $ +90^\circ $. The minus sign in the net reactance simply reflects the circuit’s capacitive nature, meaning current leads.
Step 7: Conclude the nature of the circuit
Because $X_C$ is very large compared to $X_L$ and also much larger than $R$, the circuit is predominantly capacitive.
Final Answer
The phase angle between the current and the supplied voltage is approximately $90^\circ$, and the circuit is predominantly capacitive.
