Chapter 19, Problem 11P

To determine

The four different ways for connection of resistors and equivalent resistance in each case.

Explanation of Solution

Given info:

The resistance of the resistor is, $R_1=R_2=R_3=240\Omega$ .

The three resistors may be connected as all three in series, all three in parallel, one in series and other two in parallel and one in parallel with series combination of the remaining two resistors.

Formula Used:

The expression to calculate the equivalent resistance of series connection of resistors is,$R_s=R_1+R_2+R_3$

The expression to calculate the equivalent resistance of series connection of resistors is,$R_p=\frac{R_1{R}_2{R}_3}{R_1{R}_2+R_2{R}_3+R_3{R}_1}$

The expression to calculate the equivalent resistance of one resistance in series of other two parallel resistances is,$R_{e_{12}}=R_1+\frac{R_2{R}_3}{R_2+R_3}$

The expression to calculate the equivalent resistance of one resistance in parallel of other two series resistances is,$R_{p1}=\frac{R_1\left(R_2+R_3\right)}{R_1+R_2+R_3}$

Calculation:

Substitute all the values in the above expression.

  $\begin{array}{c}{R}_s=240\Omega +240\Omega +240\Omega \\ =720\Omega \end{array}$

Substitute all the values in the above expression.

  $\begin{array}{c}{R}_p=\frac{\left(240\Omega \right)\left(240\Omega \right)\left(240\Omega \right)}{\left(240\Omega \right)\left(240\Omega \right)+\left(240\Omega \right)\left(240\Omega \right)+\left(240\Omega \right)\left(240\Omega \right)}\\ =80\Omega \end{array}$

Substitute all the values in the above expression.

  $\begin{array}{c}{R}_{e1}=240\Omega +\frac{\left(240\Omega \right)\left(240\Omega \right)}{240\Omega +240\Omega }\\ =360\Omega \end{array}$

Substitute all the values in the above expression.

  $\begin{array}{c}{R}_{p1}=\frac{\left(240\Omega \right)\left(240\Omega +240\Omega \right)}{240\Omega +240\Omega +240\Omega }\\ =160\Omega \end{array}$

Conclusion:

Therefore, the four ways of connections of resistors and equivalent resistance of each way are following:

Sr.No.	Way of connection	Equivalent resistance
1	All three in series connection	$720\Omega$
2	All three in parallel connection	$80\Omega$
3	One in series with other two in parallel connection	$360\Omega$
4	One in parallel with other two in series connection	$160\Omega$