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

Explanation

To find the time constant of the series R-C circuit connected to an AC voltage source, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Angular frequency, \( \omega = 500 \, \text{rad/s} \)
- Impedance, \( Z = R \sqrt{1.25} \)

### Step 2: Write the expression for impedance in an R-C circuit
In a series R-C circuit, the impedance \( Z \) is given by:
\[ Z = \sqrt{R^2 + X_C^2} \]
where \( X_C \) is the capacitive reactance, defined as:
\[ X_C = \frac{1}{\omega C} \]

### Step 3: Substitute \( X_C \) into the impedance formula
Substituting \( X_C \) into the impedance formula gives:
\[ Z = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} \]

### Step 4: Set up the equation using the given impedance
We know that:
\[ Z = R \sqrt{1.25} \]
Thus, we can equate the two expressions for impedance:
\[ R \sqrt{1.25} = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides leads to:
\[ R^2 \cdot 1.25 = R^2 + \left(\frac{1}{\omega C}\right)^2 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ R^2 \cdot 1.25 - R^2 = \left(\frac{1}{\omega C}\right)^2 \]
\[ 0.25 R^2 = \left(\frac{1}{\omega C}\right)^2 \]

### Step 7: Solve for \( RC \)
Taking the square root of both sides:
\[ \sqrt{0.25} R = \frac{1}{\omega C} \]
\[ 0.5 R = \frac{1}{\omega C} \]
Thus,
\[ RC = \frac{1}{0.5 \omega} \]

### Step 8: Substitute the value of \( \omega \)
Substituting \( \omega = 500 \, \text{rad/s} \):
\[ RC = \frac{1}{0.5 \times 500} = \frac{1}{250} \, \text{s} \]

### Step 9: Convert to milliseconds
To convert seconds to milliseconds:
\[ RC = \frac{1}{250} \times 1000 = 4 \, \text{ms} \]

### Step 10: State the time constant
The time constant \( \tau \) of the circuit is:
\[ \tau = RC = 4 \, \text{ms} \]

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

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