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Step-by-Step Solution
Step 1: Understand the phase difference requirement
We are given that the phase difference between the AC source voltage (emf) and the current is $ \frac{\pi}{4} $. In a series RC circuit, this phase difference (where current leads the voltage) is given by:
$ \tan(\phi) = \frac{1}{\omega RC} $
For the phase difference to be $ \frac{\pi}{4} $, we must have:
$ \tan\left(\frac{\pi}{4}\right) = 1 \quad \Longrightarrow \quad \frac{1}{\omega RC} = 1. $
Step 2: Relationship between reactance and resistance
In a series RC circuit, the capacitive reactance is:
$ X_C = \frac{1}{\omega C}. $
For a phase shift of $ \frac{\pi}{4} $, the magnitude of the reactance should match the resistance:
$ X_C = R. $
Hence,
$ \frac{1}{\omega C} = R. $
Step 3: Apply the given values to check each option
Option 1: RC circuit with R = 1 k$\Omega$ and C = 1 μF
$ R = 1000 \,\Omega $,
$ X_C = \frac{1}{\omega C} = \frac{1}{(100) \times 1 \times 10^{-6}} = 10^{4} \,\Omega. $
Here, $ X_C \neq R $ (i.e., $10^{4} \,\Omega \neq 1000 \,\Omega$), so the phase difference is not $ \frac{\pi}{4} $.
Option 2: RL circuit with R = 1 k$\Omega$ and L = 1 mH
$ R = 1000 \,\Omega, \quad
X_L = \omega L = 100 \times 10^{-3} = 0.1 \,\Omega. $
Here, $ X_L \ll R $, so the phase difference is not $ \frac{\pi}{4} $.
Option 3: RC circuit with R = 1 k$\Omega$ and C = 10 μF
$ R = 1000 \,\Omega, \quad
C = 10 \times 10^{-6} \,\text{F}, \quad
\omega = 100 \,\text{rad/s}. $
Calculate $ X_C $:
$ X_C = \frac{1}{\omega C} = \frac{1}{100 \times 10 \times 10^{-6}}
= \frac{1}{10^{-2}} = 10^{2} \times 10^{1} = 10^{3} \,\Omega. $
Thus $ X_C = 1000 \,\Omega = R $ and hence the phase difference is
$ \frac{\pi}{4} $. This matches the requirement.
Option 4: RL circuit with R = 1 k$\Omega$ and L = 10 mH
$ R = 1000 \,\Omega, \quad
X_L = \omega L = 100 \times 10 \times 10^{-3} = 1 \,\Omega. $
Clearly, $ X_L \ll R $, so the phase difference is not $ \frac{\pi}{4} $.
Step 4: Conclude the correct circuit
Only Option 3 satisfies $ X_C = R $, thereby providing a phase difference of $ \frac{\pi}{4} $. Therefore, the correct circuit is:
RC circuit with R = 1 k$\Omega$ and C = 10 μF.
