Question

A sinusoidal voltage of peak value $283\, V$ and angular frequency $320/s$ is applied to a series $LCR$ circuit. Given that $R = 5 \, \Omega , 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 :

• A. $10 \Omega $ and $\tan^{-1} \left( \frac{5}{3} \right)$
• B. $7 \Omega$ and $45^{\circ}$
• C. $10 \Omega $ and $\tan^{-1} \left( \frac{8}{3} \right)$
• D. $7 \Omega $ and $\tan^{-1} \left( \frac{5}{3} \right)$

Answer 1

The Correct Option is B

To determine the total impedance and the phase difference between the voltage across the source and the current for an \(LCR\) circuit with given values, we can follow these steps:

1. Calculate the Reactance:
   1. Inductive reactance \((X_L)\) is given by: \(X_L = \omega L\). Here, \(\omega = 320 \, \text{s}^{-1}\) and \(L = 25 \, \text{mH} = 25 \times 10^{-3} \, \text{H}\). So, 
      \(X_L = 320 \times 25 \times 10^{-3} = 8 \, \Omega\).
   2. Capacitive reactance \((X_C)\) is given by: \(X_C = \frac{1}{\omega C}\). Here, \(C = 1000 \, \mu \text{F} = 1000 \times 10^{-6} \, \text{F}\). So, 
      \(X_C = \frac{1}{320 \times 1000 \times 10^{-6}} = 3.125 \, \Omega\).
2. Determine Net Reactance: 
   Net reactance \((X)\) in the circuit is: 
   \(X = X_L - X_C = 8 - 3.125 = 4.875 \, \Omega\).
3. Calculate the Total Impedance: 
   The total impedance \((Z)\) is given by: 
   \(Z = \sqrt{R^2 + X^2}\). 
   Here, \(R = 5 \, \Omega\). Thus, 
   \(Z = \sqrt{5^2 + (4.875)^2} = \sqrt{25 + 23.7656} = \sqrt{48.7656} \approx 7 \, \Omega\).
4. Determine the Phase Difference: 
   The phase difference \(\phi\) is given by: 
   \(\phi = \tan^{-1} \left( \frac{X}{R} \right)\). 
   \(\phi = \tan^{-1} \left( \frac{4.875}{5} \right) = \tan^{-1} \left( 0.975 \right)\). 
   Solving it gives approximately 44.4 degrees, which is close to \(45^\circ\) in options.

Thus, the total impedance is \(7 \, \Omega\) and the phase difference is approximately \(45^\circ\). Therefore, the correct answer is:

\(7 \Omega\) and \(45^{\circ}\)