Chapter 35, Problem 57A

The same voltage V is impressed on each of the branches of a parallel circuit. The voltage source provides a total current $I_{toal}$

to the circuit, and ’sees' a total equivalent resistance of R in the circuit. That is, $V=I_{total}{R}_{eq}$. The total current is equal to the sum of the currents through each branch of the parallel circuit. In a circuit with n branches, $I_{toal}=I_1+I_2+I_3\mathrm{...}{I}_n$

.Use Ohm’s law $\left(I=V/R\right)$ and show how the equivalent resistance of a parallel circuit with n branches is given by the following equation.

$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\cdots +\frac{1}{R_n}$

To determine

Prove the expression $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\mathrm{...}+\frac{1}{R_n}$ .

Explanation of Solution

Given info:

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

Current in the first, second, third and nth branch of the circuit is $I_1,I_2,I_3\mathrm{...}{I}_n$ .

Resistance in first, second, third and nth branch of the circuit is $R_1,R_2,R_3\mathrm{...}{R}_n$ .

Voltage of the circuit is $V$ .

Formula Used:

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

  $I_{total}=I_1+I_2+I_3\mathrm{...}+I_n$

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 current through third resistor.

  $I_3=\frac{V}{R_3}$

Write the expression to calculate the current through nth resistance.

  $I_n=\frac{V}{R_n}$

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.

  $\begin{array}{l}\frac{V}{R_{eq}}=\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}\mathrm{...}+\frac{V}{R_n}\\ \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\mathrm{...}+\frac{1}{R_n}\end{array}$

Conclusion:

The expression of equivalent resistance $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\mathrm{...}+\frac{1}{R_n}$ is proved.