A sinusoidal voltage of peak value 283 V and angular frequency 320 s$$^{-1}$$ 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-

We begin by recalling the basic relations for a series $$LCR$$ circuit driven by an alternating voltage of angular frequency $$\omega$$.

Inductive reactance is given by the formula $$X_L=\omega L$$.

Capacitive reactance is given by the formula $$X_C=\dfrac{1}{\omega C}$$.

The net (effective) reactance is the difference of the two because the inductor and capacitor contribute opposite phasors:

$$X = X_L - X_C\;.$$

The magnitude of the total impedance is then obtained from the resistance $$R$$ and the net reactance $$X$$ via the impedance formula for a right-angled phasor triangle, i.e.

$$Z=\sqrt{R^2+X^2}\;.$$

The phase angle $$\phi$$ between the source voltage and the current follows from

$$\tan\phi=\dfrac{X}{R}\;.$$

Now we substitute the numerical values one by one.

The data are

$$R = 5\;\Omega,\;\; L = 25\text{ mH} = 25\times10^{-3}\text{ H},\;\; C = 1000\;\mu\text{F} = 1000\times10^{-6}\text{ F}=0.001\text{ F},\;\; \omega = 320\;\text{s}^{-1}.$$

First we calculate the individual reactances.

Inductive reactance:

$$X_L = \omega L = 320 \times 0.025 = 8.0\;\Omega.$$

Capacitive reactance:

$$X_C = \dfrac{1}{\omega C} = \dfrac{1}{320 \times 0.001} = \dfrac{1}{0.32} = 3.125\;\Omega.$$

Hence the net reactance is

$$X = X_L - X_C = 8.0 - 3.125 = 4.875\;\Omega.$$

Substituting $$R=5\;\Omega$$ and $$X=4.875\;\Omega$$ in the impedance expression, we have

$$Z = \sqrt{R^2 + X^2} = \sqrt{5^2 + (4.875)^2} = \sqrt{25 + 23.7656} = \sqrt{48.7656}\;\Omega.$$

Evaluating the square root,

$$Z \approx 6.99\;\Omega \approx 7\;\Omega.$$

For the phase angle, we insert the same values into $$\tan\phi=\dfrac{X}{R}$$:

$$\tan\phi = \dfrac{4.875}{5} = 0.975.$$

The arctangent of 0.975 is very close to $$45^\circ$$ (more precisely about $$44.2^\circ$$), so to the nearest whole degree we may write

$$\phi \approx 45^\circ.$$

Thus, the total impedance of the given series $$LCR$$ circuit is approximately $$7\;\Omega$$ and the phase difference between source voltage and current is about $$45^\circ$$.

Hence, the correct answer is Option B.