A series R-C combination is connected to an AC voltage of angular frequency `omega=500 radian//s`. If the impendance of the R-C circuit is `Rsqrt(1.25)`, the time constant (in millisecond) of the circuit is

Explanation

To solve the problem, we need to find the time constant of a series R-C circuit connected to an AC voltage source. The given data includes the angular frequency \( \omega = 500 \, \text{rad/s} \) and the impedance \( Z = R \sqrt{1.25} \).

### Step-by-Step Solution:

1. **Understanding Impedance in R-C Circuit**:
The impedance \( Z \) of a series R-C circuit is given by the formula:
\[ Z = \sqrt{R^2 + X_C^2} \]
where \( X_C \) is the capacitive reactance given by \( X_C = \frac{1}{\omega C} \).

2. **Substituting Given Values**:
We know that \( Z = R \sqrt{1.25} \). Therefore, we can equate the two expressions for impedance:
\[ R \sqrt{1.25} = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} \]

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ R^2 \cdot 1.25 = R^2 + \left(\frac{1}{\omega C}\right)^2 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ 1.25 R^2 - R^2 = \left(\frac{1}{\omega C}\right)^2 \]
Simplifying this results in:
\[ 0.25 R^2 = \left(\frac{1}{\omega C}\right)^2 \]

5. **Expressing \( C \)**:
From the equation above, we can express \( C \):
\[ \left(\frac{1}{\omega C}\right)^2 = 0.25 R^2 \]
Taking the square root:
\[ \frac{1}{\omega C} = 0.5 R \]
Therefore:
\[ C = \frac{1}{0.5 R \omega} = \frac{2}{R \omega} \]

6. **Finding the Time Constant**:
The time constant \( \tau \) of an R-C circuit is given by:
\[ \tau = R \cdot C \]
Substituting \( C \):
\[ \tau = R \cdot \frac{2}{R \omega} = \frac{2}{\omega} \]

7. **Substituting the Value of \( \omega \)**:
Now substituting \( \omega = 500 \, \text{rad/s} \):
\[ \tau = \frac{2}{500} = 0.004 \, \text{seconds} \]

8. **Converting to Milliseconds**:
To convert seconds to milliseconds:
\[ \tau = 0.004 \times 1000 = 4 \, \text{milliseconds} \]

### Final Answer:
The time constant of the circuit is \( \tau = 4 \, \text{milliseconds} \).