In an ac circuit , an alternating voltage `e = 200 sqrt2 sin 100 t` volts is connected to a capacitor of capacitance `1 mu F`. The rms value of the current in the circuit is :

Options

A. 20 mA
B. 10 mA
C. 100 mA
D. 200 mA

Answer

To find the RMS value of the current in the given AC circuit, we can follow these steps:

### Step 1: Identify the given parameters
The alternating voltage is given as:
\[ e(t) = 200 \sqrt{2} \sin(100 t) \text{ volts} \]
The capacitance of the capacitor is:
\[ C = 1 \mu F = 1 \times 10^{-6} F \]

### Step 2: Determine the angular frequency (ω)
From the voltage equation, we can identify the angular frequency (ω):
\[ \omega = 100 \text{ rad/s} \]

### Step 3: Calculate the RMS value of the voltage (E_rms)
The RMS value of the voltage is given by the formula:
\[ E_{rms} = \frac{E_0}{\sqrt{2}} \]
Where \( E_0 = 200 \sqrt{2} \).
Substituting the value:
\[ E_{rms} = \frac{200 \sqrt{2}}{\sqrt{2}} = 200 \text{ volts} \]

### Step 4: Calculate the capacitive reactance (X_c)
The capacitive reactance (X_c) is given by the formula:
\[ X_c = \frac{1}{\omega C} \]
Substituting the values:
\[ X_c = \frac{1}{100 \times 1 \times 10^{-6}} = \frac{1}{10^{-4}} = 10^4 \text{ ohms} \]

### Step 5: Calculate the RMS value of the current (I_rms)
The RMS value of the current can be calculated using the formula:
\[ I_{rms} = \frac{E_{rms}}{X_c} \]
Substituting the values:
\[ I_{rms} = \frac{200}{10^4} = 0.02 \text{ A} \]
Converting to milliamperes:
\[ I_{rms} = 20 \text{ mA} \]

### Final Answer:
The RMS value of the current in the circuit is:
\[ I_{rms} = 20 \text{ mA} \]

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