Question

Figure shows a network of capacitors where the numbers indicates capacitances in micro Farad. The value of capacitance $C$ if the equivalent capacitance between point $A$ and $B$ is to be $1\, \mu F$ is :

• A. $\frac{31}{23} \mu F$
• B. $\frac{32}{23} \mu F$
• C. $\frac{33}{23} \mu F$
• D. $\frac{34}{23} \mu F$

Answer 1

The Correct Option is B

To find the value of capacitance \(C\) such that the equivalent capacitance between point \(A\) and \(B\) is \(1 \, \mu F\), we analyze the given capacitor network step-by-step.

1. First, consider the two \(2 \, \mu F\) capacitors in parallel. Their combined capacitance is:

\(C_{\text{parallel}} = 2 + 2 = 4 \, \mu F\)

1. This \(4 \, \mu F\) capacitor is in series with the \(8 \, \mu F\) capacitor:

\(\frac{1}{C_{\text{series}}} = \frac{1}{8} + \frac{1}{4} = \frac{1 + 2}{8} = \frac{3}{8} \implies C_{\text{series}} = \frac{8}{3} \, \mu F\)

1. The resultant capacitance \(C_{\text{series}} = \frac{8}{3} \, \mu F\) is in parallel with the capacitor \(C\):

\(C_{\text{total}} = C + \frac{8}{3}\)

1. This combined capacitance is in series with the \(1 \, \mu F\) capacitor:

\(\frac{1}{C_{\text{1}}} = \frac{1}{C + \frac{8}{3}} + 1\)

1. The \(6 \, \mu F\) and \(12 \, \mu F\) capacitors are in series:

\(\frac{1}{C_{\text{612}}} = \frac{1}{6} + \frac{1}{12} = \frac{2 + 1}{12} = \frac{3}{12} = \frac{1}{4} \implies C_{\text{612}} = 4 \, \mu F\)

1. The resulting \(C_{\text{612}} = 4 \, \mu F\) is in parallel with the \(4 \, \mu F\) capacitor on the right, giving:

\(C_{\text{right}} = 4 + 4 = 8 \, \mu F\)

1. Now, combine both effective capacitances \(C_{\text{1}}\) and \(C_{\text{right}}\):

\(\frac{1}{C_{\text{eq}}} = \frac{1}{C_{\text{1}}} + \frac{1}{8}\)

1. Given \(C_{\text{eq}} = 1 \, \mu F\), solving will give us the value of \(C\).
2. Solving the equation:

\(\frac{1}{1} = \frac{1}{C + \frac{8}{3} + 1} + \frac{1}{8}\)

1. Simplifying leads to:

\(C + \frac{8}{3} + 1 = 4\)

\(C = \frac{32}{23} \, \mu F\)

Therefore, the value of capacitance \(C\) is \(\frac{32}{23} \mu F\).