A sinusoidal voltage of peak value 283 V and angular frequency 320/s is applied to a series LCR circuit.Given that `R = 5Omega, L = 25 mH` and `C = 1000 mu F`.The total impedance, and phase difference between the voltage across the source and the current will respectively be:

Options

A. `10 Omega and tan^(-1)((5)/(3))`
B. `7 Omega and 45^(@)`
C. `10 Omega and tan^(-1)((8)/(3))`
D. `7 Omega and tan^(-1)((5)/(3))`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the Inductive Reactance (XL)
The inductive reactance (XL) is given by the formula:
\[ X_L = \omega L \]
Where:
- \( \omega = 320 \, \text{s}^{-1} \) (angular frequency)
- \( L = 25 \, \text{mH} = 25 \times 10^{-3} \, \text{H} \)

Now substituting the values:
\[ X_L = 320 \times 25 \times 10^{-3} \]
\[ X_L = 8 \, \Omega \]

### Step 2: Calculate the Capacitive Reactance (XC)
The capacitive reactance (XC) is given by the formula:
\[ X_C = \frac{1}{\omega C} \]
Where:
- \( C = 1000 \, \mu\text{F} = 1000 \times 10^{-6} \, \text{F} \)

Now substituting the values:
\[ X_C = \frac{1}{320 \times 1000 \times 10^{-6}} \]
\[ X_C = \frac{1}{0.32} \]
\[ X_C = 3.125 \, \Omega \]

### Step 3: Calculate the Total Impedance (Z)
The total impedance (Z) in a series LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Where:
- \( R = 5 \, \Omega \)

Substituting the values:
\[ Z = \sqrt{5^2 + (8 - 3.125)^2} \]
\[ Z = \sqrt{25 + (4.875)^2} \]
\[ Z = \sqrt{25 + 23.765625} \]
\[ Z = \sqrt{48.765625} \]
\[ Z \approx 6.98 \, \Omega \approx 7 \, \Omega \]

### Step 4: Calculate the Phase Difference (φ)
The phase difference (φ) between the voltage and the current is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]

Substituting the values:
\[ \tan(\phi) = \frac{8 - 3.125}{5} \]
\[ \tan(\phi) = \frac{4.875}{5} \]
\[ \tan(\phi) = 0.975 \]

Now calculating φ:
\[ \phi = \tan^{-1}(0.975) \]
Using a calculator:
\[ \phi \approx 44.27^\circ \approx 45^\circ \]

### Final Result
The total impedance \( Z \) is approximately \( 7 \, \Omega \) and the phase difference \( \phi \) is approximately \( 45^\circ \).

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