Chapter 25, Problem 76P

(a)

To determine

The equivalent resistance between $a$ and $b$ .

Answer to Problem 76P

The equivalent resistance of the circuit is $4.10\Omega$ .

Explanation of Solution

Formula used:

The expression for equivalent resistance in upper wire is given by,

  $R_{\text{eq,1}}=R_6+\frac{\left(R_2+R_4\right)\left(R_4\right)}{\left(R_2+R_4\right)+R_4}$

The expression for equivalent resistance in lower wire for is given by,

  $R_{\text{eq,2}}=\frac{R_8^2}{2R_8}+R_4$

The expression for net equivalent resistance is given by,

  $R_{\text{eq}}=\frac{R_{\text{eq,1}}{R}_{\text{eq,2}}}{R_{\text{eq,1}}+R_{\text{eq,2}}}$

Calculation:

The equivalent resistance in upper wire is calculated as,

  $\begin{array}{c}{R}_{\text{eq,1}}=R_6+\frac{\left(R_2+R_4\right)\left(R_4\right)}{\left(R_2+R_4\right)+R_4}\\ =6\Omega +\frac{\left(2\Omega +4\Omega \right)\left(4\Omega \right)}{\left(2\Omega +4\Omega \right)+\left(4\Omega \right)}\\ =8.4\Omega \end{array}$

The equivalent resistance in lower wire is calculated as,

  $\begin{array}{c}{R}_{\text{eq,2}}=\frac{R_8^2}{2R_8}+R_4\\ =\frac{{\left(8\Omega \right)}^2}{2\left(8\Omega \right)}+4\Omega \\ =4\Omega +4\Omega \\ =8\Omega \end{array}$

The net equivalent resistance of the circuit is calculated as,

  $\begin{array}{c}{R}_{\text{eq}}=\frac{R_{\text{eq,1}}{R}_{\text{eq,2}}}{R_{\text{eq,1}}+R_{\text{eq,2}}}\\ =\frac{\left(8.4\Omega \right)\left(8\Omega \right)}{\left(8.4\Omega \right)+\left(8\Omega \right)}\\ =4.10\Omega \end{array}$

Conclusion:

Therefore, the equivalent resistance of the circuit is $4.10\Omega$ .

(b)

To determine

The current in each resistor.

Answer to Problem 76P

The currents through upper wire $6\Omega$ resistor, lower wire $4\Omega$ resistor, $4\Omega$ resistor parallel with $2\Omega$ and $4\Omega$ resistors, upper wire $6\Omega$

  $=\left(4\Omega +2\Omega \right)$ resistor and, lower wire in parallel $8\Omega$ resistor are $1.43\text{A}$ , $1.50\text{A}$ , $0.86\text{A}$ , $0.57\text{A}$ and $0.75\text{A}$ respectively.

Explanation of Solution

Formula used:

The expression for current in upper wire $6\Omega$ resistor is given by,

  $I_6=\frac{V}{R_{\text{eq,1}}}$

The expression for current in upper wire $4\Omega$ resistor is given by,

  $I_{\text{4}}=\frac{V-\left(I_6\right)\left(R_6\right)}{R_{\text{4}}}$

The expression for current in upper wire $6\Omega$ resistor is given by,

  $I_{6,\text{upper}}=\frac{V-\left(I_6\right)\left(R_6\right)}{R_{\text{6}}}$

The expression for current in lower wire is given by,

  $I_{\text{lower}}=\frac{V}{R_{\text{eq,2}}}$

The expression for current in lower wire in parallel $8\Omega$ resistor is given by,

  $I_{\text{8}}=\frac{V-\left(I_{\text{lower}}\right)\left(R_4\right)}{R_8}$

Calculation:

The current in upper wire $6\Omega$ resistor is calculated as,

  $\begin{array}{c}{I}_6=\frac{V}{R_{\text{eq,1}}}\\ =\frac{12.0\text{V}}{8.4\Omega }\\ =1.43\text{A}\end{array}$

The current in upper wire $4\Omega$ resistor is calculated as,

  $\begin{array}{c}{I}_{\text{4}}=\frac{V-\left(I_6\right)\left(R_6\right)}{R_{\text{4}}}\\ =\frac{12.0\text{V}-\left(1.43\text{A}\right)\left(6\Omega \right)}{4\Omega }\\ =0.86\text{A}\end{array}$

The current in upper wire $6\Omega$ resistor is calculated as,

  $\begin{array}{c}{I}_{6,\text{upper}}=\frac{V-\left(I_6\right)\left(R_6\right)}{R_{\text{6}}}\\ =\frac{12.0\text{V}-\left(1.43\text{A}\right)\left(6\Omega \right)}{6\Omega }\\ =0.57\text{A}\end{array}$

The current in lower wire is calculated as,

  $\begin{array}{c}{I}_{\text{lower}}=\frac{V}{R_{\text{eq,2}}}\\ =\frac{12.0\text{V}}{8\Omega }\\ =1.5\text{A}\end{array}$

The current in lower wire in parallel $8\Omega$ resistor is calculated as,

  $\begin{array}{c}{I}_{\text{8}}=\frac{V-\left(I_{\text{lower}}\right)\left(R_4\right)}{R_8}\\ =\frac{12.0\text{V}-\left(1.5\text{A}\right)\left(4\Omega \right)}{8\Omega }\\ =0.75\text{A}\end{array}$

Conclusion:

Therefore, the currents through upper wire $6\Omega$ resistor, lower wire $4\Omega$ resistor, $4\Omega$ resistor parallel with $2\Omega$ and $4\Omega$ resistors, upper wire $6\Omega$ $=\left(4\Omega +2\Omega \right)$ resistor and, lower wire in parallel $8\Omega$ resistor are $1.43\text{A}$ , $1.50\text{A}$ , $0.86\text{A}$ , $0.57\text{A}$ and $0.75\text{A}$ respectively.