Chapter 17, Problem 95QAP

To determine

All the unique combinations using five identical capacitors of $10\mu F$ each.

Explanation of Solution

The following are the combinations
(1)

  
  $\begin{array}{l}\frac{1}{C_{equiv}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{5}{C}\\ C_{equiv}=\frac{C}{5}=\frac{10\mu F}{5}=2\mu F\end{array}$

(2)

  
  $\begin{array}{l}\frac{1}{C_{top}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{4}{C}\\ C_{equiv}=\frac{C}{4}+C=\frac{5C}{4}=\frac{5(10\mu F)}{4}=13\mu F\end{array}$

(3)

  
  $\begin{array}{l}\frac{1}{C_{top}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{3}{C}\\ \frac{1}{C_{bottom}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\\ C_{equiv}=C_{top}+C_{bottom}=\frac{C}{3}+\frac{C}{2}=\frac{5C}{6}=\frac{5(10\mu F)}{6}=8.3\mu F\end{array}$

(4)

  
  $\begin{array}{l}\frac{1}{C_{top}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\\ \frac{1}{C_{bottom}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\\ C_{parallel}=C_{top}+C_{bottom}=\frac{C}{2}+\frac{C}{2}=C\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\\ \text{}{C}_{equiv}=\frac{C}{2}=5\mu F\end{array}$

(5)

  
  $\begin{array}{l}\frac{1}{C_{top}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{3}{C}\\ \frac{1}{C_{bottom}}=\frac{1}{C}\\ C_{parallel}=C_{top}+C_{bottom}=\frac{C}{3}+C=\frac{4C}{3}\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}=\frac{3}{4C}+\frac{1}{C}=\frac{7}{4C}\\ \text{}{C}_{equiv}=\frac{4C}{7}=5.7\mu F\end{array}$

(6)

  
  $\begin{array}{l}\frac{1}{C_{top}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\\ C_{parallel}=C_{top}+C=\frac{C}{2}+C=\frac{3C}{2}\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}+\frac{1}{C}=\frac{2}{3C}+\frac{1}{C}+\frac{1}{C}=\frac{8}{3C}\\ \text{}{C}_{equiv}=\frac{3C}{8}=3.8\mu F\end{array}$

(7)

  
  $\begin{array}{l}{C}_{parallel}=C+C=2C\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{1}{2C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{7}{2C}\\ \text{}{C}_{equiv}=\frac{2C}{7}=2.9\mu F\end{array}$

(8)

  
  $\begin{array}{l}{C}_{parallel}=C+C+C=3C\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}+\frac{1}{C}=\frac{1}{3C}+\frac{1}{C}+\frac{1}{C}=\frac{7}{3C}\\ \text{}{C}_{equiv}=\frac{3C}{7}=4.3\mu F\end{array}$

(9)

  
  $\begin{array}{l}{C}_{parallel,1}=C+C+C=3C\\ C_{parallel,2}=C+C=2C\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel,1}}+\frac{1}{C_{parallel,2}}=\frac{1}{3C}+\frac{1}{2C}=\frac{5}{6C}\\ \text{}{C}_{equiv}=\frac{6C}{5}=12\mu F\end{array}$

(10

  
  $\begin{array}{l}{C}_{parallel}=C+C+C+C=4C\\ \frac{1}{C_{equiv}}=\frac{1}{C_{parallel}}+\frac{1}{C}=\frac{1}{4C}+\frac{1}{C}=\frac{5}{4C}\\ \text{}{C}_{equiv}=\frac{4C}{5}=8\mu F\end{array}$

(11)

  
  $C_{equiv}=C+C+C+C+C=5C=50\mu F$