Question:
A \(100\,\mu\text{F}\) capacitor is connected to a \(100\,\text{V}\), \(50\,\text{Hz}\) AC supply. The rms value of the current is:
A \(100\,\mu\text{F}\) capacitor is connected to a \(100\,\text{V}\), \(50\,\text{Hz}\) AC supply. The rms value of the current is:
Show Hint
For a pure capacitor in an AC circuit,
\[
X_C=\frac{1}{2\pi fC}
\]
and
\[
I_{\text{rms}}=\frac{V_{\text{rms}}}{X_C}.
\]
Also, current leads voltage by \(90^\circ\) in a purely capacitive circuit.
Updated On: Jun 26, 2026
- \(3.14\,\text{A}\)
- \(4.75\,\text{A}\)
- \(2.33\,\text{A}\)
- \(5.5\,\text{A}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Calculate the capacitive reactance.
The capacitive reactance is given by \[ X_C=\frac{1}{2\pi fC} \] Given, \[ f=50\,\text{Hz} \] \[ C=100\,\mu\text{F}=100\times 10^{-6}\,\text{F} \] Substituting the values, \[ X_C=\frac{1}{2\pi(50)(100\times 10^{-6})} \] \[ X_C=\frac{1}{\pi\times 10^{-2}} \] \[ X_C=\frac{100}{\pi} \] \[ X_C\approx 31.8\,\Omega \]
Step 2: Use the AC current relation.
The rms current in a purely capacitive AC circuit is \[ I_{\text{rms}}=\frac{V_{\text{rms}}}{X_C} \] Given, \[ V_{\text{rms}}=100\,\text{V} \] Therefore, \[ I_{\text{rms}} = \frac{100}{100/\pi} \] \[ I_{\text{rms}}=\pi \] \[ I_{\text{rms}}\approx 3.14\,\text{A} \]
Step 3: Verify with the options.
The calculated current is \[ 3.14\,\text{A} \] which matches option (1).
Step 4: Final conclusion.
Therefore, the rms current is \[ \boxed{3.14\,\text{A}} \]
The capacitive reactance is given by \[ X_C=\frac{1}{2\pi fC} \] Given, \[ f=50\,\text{Hz} \] \[ C=100\,\mu\text{F}=100\times 10^{-6}\,\text{F} \] Substituting the values, \[ X_C=\frac{1}{2\pi(50)(100\times 10^{-6})} \] \[ X_C=\frac{1}{\pi\times 10^{-2}} \] \[ X_C=\frac{100}{\pi} \] \[ X_C\approx 31.8\,\Omega \]
Step 2: Use the AC current relation.
The rms current in a purely capacitive AC circuit is \[ I_{\text{rms}}=\frac{V_{\text{rms}}}{X_C} \] Given, \[ V_{\text{rms}}=100\,\text{V} \] Therefore, \[ I_{\text{rms}} = \frac{100}{100/\pi} \] \[ I_{\text{rms}}=\pi \] \[ I_{\text{rms}}\approx 3.14\,\text{A} \]
Step 3: Verify with the options.
The calculated current is \[ 3.14\,\text{A} \] which matches option (1).
Step 4: Final conclusion.
Therefore, the rms current is \[ \boxed{3.14\,\text{A}} \]
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