A series CR circuit with $\displaystyle \mathrm{R}=200 \Omega$ and $\displaystyle \mathrm{C}=(50 / \pi) \mu \mathrm{F}$ is connected across an ac source of peak voltage $\displaystyle \varepsilon_{0}=100 \mathrm{~V}$ and frequency $\displaystyle v=50 \mathrm{~Hz}$. Calculate

(a)

impedance of the circuit (Z), (b) phase angle ( $\displaystyle \phi$ ), and (c) voltage across the resistor.

Calculating\[\text { (a) } \begin{aligned} & Z=\sqrt{\mathrm{R}^{2}+X_{c}^{2}}==\sqrt{\mathrm{R}^{2}+\left(\frac{1}{2 \pi v \mathrm{C}}\right)^{2}} \\ & X_{c}=\frac{1}{2 \pi v \mathrm{C}}=\frac{1}{2 \times \pi \times 50 \times \dfrac{50}{\pi} \times 10^{-6}}=200 \Omega \\ & Z=\sqrt{(200)^{2}+(200)^{2}}=200 \sqrt{2} \Omega \\ & \approx 282 \Omega \end{aligned} \] \[\text { (b) } \begin{aligned} \tan \phi & =\frac{X_{c}}{\mathrm{R}}=\frac{200}{200} \\ \phi & =45^{0} \quad \text { or } \frac{\pi}{4} \mathrm{rad} \end{aligned} \] \[\text { (c) } \begin{aligned} V_{r m s} & =I_{r m s} \mathrm{R}=\frac{V_{r m s}}{Z} \mathrm{R} \\ & =\frac{100}{\sqrt{2} \times 200 \sqrt{2}} \times 200 \\ & =50 \mathrm{~V} \end{aligned} \]