Chapter 35, Problem 58A

Show how the expression given in Question 57, when applied to two branches with resistances $R_1$ and $R_2$, can be written as follows.

$R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$

To determine

Prove the expression $R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$ .

Answer to Problem 58A

The expression of equivalent resistance $R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$ is proved.

Explanation of Solution

Given info:

Total current in the circuit is $I_{total}$ .

Current in the first, second, branch of the circuit is $I_1,I_2$ .

Resistance in first, second, branch of the circuit is, $R_1,R_2$ .

Voltage of the circuit is, $V$

Formula Used:

Write the expression to calculate the total current of the circuit.

  $I_{total}=I_1+I_2$

Write the expression to calculate the current through first resistor.

  $I_1=\frac{V}{R_1}$

Write the expression to calculate the current through second resistor.

  $I_2=\frac{V}{R_2}$

Write the expression to calculate the total current through circuit.

  $I_{total}=\frac{V}{R_{eq}}$

Calculation:

Substitute the values of currents through resistors in the circuit in the expression of total current through circuit.

  $\begin{array}{c}\frac{V}{R_{eq}}=\frac{V}{R_1}+\frac{V}{R_2}\\ \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}\\ \frac{1}{R_{eq}}=\frac{R_2+R_1}{R_1{R}_2}\\ R_{eq}=\frac{R_1{R}_2}{R_1+R_2}\end{array}$

Conclusion:

The expression of equivalent resistance $R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$ is proved.