Question

Three resistors in parallel have an equivalent resistance of 15 ? . Two of the resistors have resistances of 40 ? and 60 ? .

What is the resistance of the third resistor?

Answer 1

Concepts and reason

The concept that is used to solve this problem is the equivalent resistance of the parallel combination of resistances.

Calculate the value of unknown resistance by using the expression of equivalent resistance of the parallel combination of resistance.

Fundamentals

Write the expression for the equivalent resistance for the parallel combination of resistance.

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

Here, ${R_{eq}}$ is the total equivalent resistance of the resistors, and${R_1}$, ${R_2}$ and ${R_3}$ are the parallel connected resistors.

The expression of unknown resistance is determined by the using the relation of the equivalent resistance for the parallel combination of resistance.

The expression for the equivalent resistance for the parallel combination of resistance is,

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

Here, ${R_{eq}}$ is the total equivalent resistance, ${R_1}$, ${R_2}$ and ${R_3}$are the resistors.

Rearrange the expression for unknown resistance${R_3}$.

$\frac{1}{{{R_3}}} = \frac{1}{{{R_{eq}}}} - \frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}$ …… (1)

Substitute ${\rm{15 }}\Omega$ for ${R_{eq}}$,${\rm{40 }}\Omega$ for ${R_1}$, and ${\rm{60 }}\Omega$ for ${R_2}$ in equation (1).

$\begin{array}{c}\\\frac{1}{{{R_3}}} = \frac{1}{{{\rm{15 }}\Omega }} - \frac{1}{{40\;\Omega }} - \frac{1}{{60\;\Omega }}\\\\ = \frac{1}{{40\;\Omega }}\\\end{array}$

Solve for ${R_3}$.

${R_3} = 40{\rm{ }}\Omega$

Ans:

The value of unknown resistance is $40{\rm{ }}\Omega$.