$$\text { Figure } 10

Question

$$\text { Figure } 10 < 1$$
-If the frequency equals  60 Hz  in Figure  10-1 , what is the value of capacitance?

Quizplus · Accepted Answer

From Figure 10-1, we can see that:
\begin{align*}
Z &= \sqrt{R^2 + (X_L - X_C)^2} \
X_L - X_C &= 2\pi f L - \frac{1}{2\pi f C} \
&= 2\pi \cdot 60 \cdot 0.1 - \frac{1}{2\pi \cdot 60 \cdot C}\
&= 12\pi - \frac{1}{360\pi C}
\end{align*}
Since the impedance Z is purely resistive, we have:
\begin{align*}
Z = R &= 100 \
\Rightarrow C &= \frac{1}{360\pi \cdot (12\pi - X_L + X_C)} \
&= \frac{1}{360\pi \cdot (12\pi - (2\pi \cdot 60 \cdot 0.1 + X_C))} \
&= \frac{1}{360\pi(10\pi - X_C)}
\end{align*}
Substituting the value of R and simplifying:
\begin{align*}
100 &= \frac{1}{360\pi(10\pi - X_C)} \
\Rightarrow X_C &= 10\pi - \frac{1}{36\pi} \
&\approx 31.4 \mathrm{\Omega}
\end{align*}
Finally, using the formula for capacitive reactance:
\begin{align*}
X_C &= \frac{1}{2\pi f C} \
\Rightarrow C &= \frac{1}{2\pi f X_C} \
&= \frac{1}{2\pi \cdot 60 \cdot 31.4} \
&\approx 22 \mu \mathrm{F}
\end{align*}
Therefore, the answer is B.

Figure 10<1\text { Figure } 10 < 1 Figure 10<1 -If the Frequency Equals 60 Hz in Figure

$\text { Figure } 10 < 1$ -If the Frequency Equals 60 Hz in Figure