Question

Three capacitances 3\(\mu\)F, 9\(\mu\)F, and 18\(\mu\)F are connected once in series and then in parallel. The ratio of equivalent Cs/Cр will be:

• A. 15:1
• B. 1:1
• C. 1:3
• D. 1:15

Answer 1

The Correct Option is D

To determine the ratio of the equivalent capacitance when the capacitors are connected in series (\(C_s\)) and in parallel (\(C_p\)), follow these steps:

Step 1: Calculate Equivalent Capacitance in Series:

For capacitors connected in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of their individual capacitances.

\[\frac{1}{C_s}=\frac{1}{3}+\frac{1}{9}+\frac{1}{18}\]

Simplifying:

\[\frac{1}{C_s}=\frac{6}{18}=\frac{1}{3}\]

Thus, \(C_s = 3\)µF.

Step 2: Calculate Equivalent Capacitance in Parallel:

For capacitors connected in parallel, the equivalent capacitance is the sum of the individual capacitances.

\[C_p = 3 + 9 + 18 = 30\]µF

Step 3: Calculate Ratio \(C_s/C_p\):

The ratio of the equivalent capacitance in series to parallel is:

\[\frac{C_s}{C_p}=\frac{3}{30}=\frac{1}{10}\]

We need to adjust \(C_s\) as there was an oversight. Let's recalculate the series correctly:

\[\frac{1}{C_s}=\frac{1}{3}+\frac{1}{9}+\frac{1}{18}=\frac{6+2+1}{18}=\frac{9}{18}=\frac{1}{2}\]

Thus, \(C_s=2\)µF.

Recalculating the ratio:

\[\frac{C_s}{C_p}=\frac{2}{30}=\frac{1}{15}\]

Therefore, the final ratio is \(1:15\).