Time Constant of Series RC Circuit in AC Source

For a series $R$-$C$ circuit connected to an AC source of angular frequency $\omega$, the impedance $Z$ is given by:
$Z = \sqrt{R^2 + X_C^2}$

where $X_C$ is the capacitive reactance:
$X_C = \frac{1}{\omega C}$

Given that the impedance of the circuit is $Z = R\sqrt{1.25}$, we can write:
$Z^2 = R^2 + X_C^2$
$(R\sqrt{1.25})^2 = R^2 + X_C^2$
$1.25 R^2 = R^2 + X_C^2$
$X_C^2 = 0.25 R^2$

Taking the square root on both sides:
$X_C = 0.5 R = \frac{R}{2}$

Substituting the expression for capacitive reactance, $X_C = \frac{1}{\omega C}$:
$\frac{1}{\omega C} = \frac{R}{2}$
$RC = \frac{2}{\omega}$

The time constant $\tau$ of an $R$-$C$ circuit is defined as $\tau = RC$. Given $\omega = 500 \text{ rad/s}$:
$\tau = \frac{2}{500}\text{ s} = 4 \times 10^{-3}\text{ s} = 4\text{ ms}$

Thus, the time constant of the circuit is $4\text{ ms}$.