A sinusoidal supply of frequency 10 Hz and r.m.s. voltage 12 V is connected to a 2.1 `muF` capacitor. What is r.m.s. value of current?

Options

A. 5.5 mA
B. 20 mA
C. 26 mA
D. 1.6 mA

Answer

To find the r.m.s. value of current in the given AC circuit with a capacitor, we can follow these steps:

### Step 1: Identify Given Values
- Frequency (f) = 10 Hz
- R.M.S. Voltage (Vrms) = 12 V
- Capacitance (C) = 2.1 µF = 2.1 × 10^(-6) F

### Step 2: Calculate Angular Frequency (ω)
The angular frequency (ω) is calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 10 \]
\[ \omega = 20\pi \, \text{rad/s} \]

### Step 3: Calculate Capacitive Reactance (Xc)
The capacitive reactance (Xc) is given by:
\[ X_c = \frac{1}{\omega C} \]
Substituting the values of ω and C:
\[ X_c = \frac{1}{20\pi \times 2.1 \times 10^{-6}} \]

Calculating this:
\[ X_c \approx \frac{1}{20 \times 3.14 \times 2.1 \times 10^{-6}} \]
\[ X_c \approx \frac{1}{1.319 \times 10^{-4}} \]
\[ X_c \approx 7577.23 \, \Omega \]

### Step 4: Calculate R.M.S. Current (Irms)
The r.m.s. current (Irms) can be calculated using the formula:
\[ I_{rms} = \frac{V_{rms}}{X_c} \]
Substituting the values:
\[ I_{rms} = \frac{12}{7577.23} \]

Calculating this:
\[ I_{rms} \approx 0.00158 \, \text{A} \]
\[ I_{rms} \approx 1.58 \, \text{mA} \]

### Final Answer
The r.m.s. value of current is approximately **1.58 mA**.

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