Chapter 19, Problem 83GP

To determine

The value of the resistor $R$ .

Answer to Problem 83GP

The value of the resistor $R$ is $100\Omega$ .

Explanation of Solution

Given information:

Consider the given circuit.

Formula used:

Apply Kirchhoff’s voltage law to calculate the resistance.

Calculation:

Consider the circuit when switches are open.

Since the switches are open. So, the current will not flow across resistor $R$ . Apply Ohm’s law to calculate the current flowing through the battery.

  $I=\frac{6}{\begin{array}{c}20+50+10\\ =0.075\text{A}\end{array}}$

Calculate the voltage across $20\Omega$ resistor using Ohm’s law.

  $\begin{array}{c}{V}_{20}=20\times 0.075\\ =1.5\text{V}\end{array}$

Consider the circuit when switches are closed.

Apply Kirchhoff’s voltage law in the outer loop.

  $\begin{array}{c}-6+50I_1+1.5=0\\ I_1=\frac{6-1.5}{50}\\ =0.09\text{A}\end{array}$

Calculate the current flowing through resistor $R$ .

  $\begin{array}{c}{I}_R=0.09-0.075\\ =0.015\text{A}\end{array}$

Apply Ohm’s law to calculate the resistance $R$ .

  $\begin{array}{c}R=\frac{1.5}{0.015}\\ =100\Omega \end{array}$

Therefore, the value of the resistor $R$ is $100\Omega$ .