A 10 `Omega` resistor is connected across an alternating power supply of 110 V. Calculate the time taken by the current to reach the root mean square value of current from its maximum value . The frequency of the source is 50 Hz.

Answer

To solve the problem, we need to calculate the time taken by the current to reach its root mean square (RMS) value from its maximum value in a purely resistive circuit connected to an AC power supply.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Resistance (R) = 10 Ω
- RMS Voltage (V_rms) = 110 V
- Frequency (f) = 50 Hz

2. **Calculate the Maximum Voltage (V₀):**
The relationship between RMS voltage and maximum voltage is given by:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]
Rearranging gives:
\[ V_0 = V_{rms} \times \sqrt{2} = 110 \, \text{V} \times \sqrt{2} \approx 155.56 \, \text{V} \]

3. **Calculate the Maximum Current (I₀):**
Using Ohm's Law, the maximum current can be calculated as:
\[ I_0 = \frac{V_0}{R} = \frac{155.56 \, \text{V}}{10 \, \Omega} \approx 15.56 \, \text{A} \]

4. **Determine the Angular Frequency (ω):**
The angular frequency is calculated using:
\[ \omega = 2\pi f = 2\pi \times 50 \, \text{Hz} = 100\pi \, \text{rad/s} \]

5. **Relate RMS Current to Maximum Current:**
The RMS current (I_rms) is related to the maximum current by:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} = \frac{15.56 \, \text{A}}{\sqrt{2}} \approx 11 \, \text{A} \]

6. **Find the Time to Reach RMS Current:**
The current in a purely resistive AC circuit can be expressed as:
\[ I(t) = I_0 \sin(\omega t) \]
We need to find the time \( t \) when the current reaches the RMS value:
\[ I_{rms} = I_0 \sin(\omega t) \]
Substituting the values:
\[ 11 = 15.56 \sin(100\pi t) \]
Rearranging gives:
\[ \sin(100\pi t) = \frac{11}{15.56} \approx 0.707 \]

7. **Calculate the Angle:**
The angle corresponding to \( \sin^{-1}(0.707) \) is:
\[ 100\pi t = \frac{\pi}{4} \]

8. **Solve for Time (t):**
\[ t = \frac{\pi/4}{100\pi} = \frac{1}{400} \, \text{s} = 0.0025 \, \text{s} = 2.5 \, \text{ms} \]

### Final Answer:
The time taken by the current to reach the root mean square value from its maximum value is **2.5 milliseconds**.