Question

The currents in two resistors in a circuit are shown in the figure above.

What is the value of resistor R?

Answer 1

Concepts and reason

The concepts used to solve this problem are Ohm’s law in terms of current and resistance in the circuit and Kirchhoff’s laws.

Initially, the current through the unknown resistance can be calculated by using Kirchhoff’s junction rule. Later, by using the concept of same potential drop for a parallel and series connection, the unknown resistance can be calculated.

Fundamentals

Ohm’s law states that “the potential difference between two points in a wire is proportional to the current in the wire”.

According to Ohm’s law,

$V = IR$

Here, $R$ is the resistance and $I$ is the current.

Kirchhoff’s law states that “the total current entering a junction is equal to the total current leaving the junction”

The expression for the current through the unknown resistance is as follows:

${I_R} = \left( {{I_2} - {I_1}} \right)$

Here, ${I_R}$ is the current through the unknown resistance, ${I_1}$ and ${I_2}$ are the given currents.

Substitute $2.5\;{\rm{A}}$ for ${I_2}$ and $1.5\;{\rm{A}}$ for ${I_1}$ .

$\begin{array}{c}\\{I_R} = \left[ {\left( {2.5\;{\rm{A}}} \right) - \left( {1.5\;{\rm{A}}} \right)} \right]\\\\ = 1\;{\rm{A}}\\\end{array}$

According to Ohm’s law for the known resistance:

$V = {I_1}{R_1}$

Here, ${R_1}$ is the resistance connected in parallel with unknown resistance, and V is the potential.

According to Ohm’s law for the unknown resistance:

$V = {I_R}R$

By comparing the above two equations, the expression for the unknown resistance is as follows:

${I_R}R = {I_1}{R_1}$

Substitute $1\;{\rm{A}}$ for ${I_R}$ , $1.5\;{\rm{A}}$ for ${I_1}$ , and $200\;\Omega$ for ${R_1}$ to find $R$ .

$\begin{array}{c}\\\left( {1\;{\rm{A}}} \right)R = \left( {1.5\;{\rm{A}}} \right)\left( {200\;\Omega } \right)\\\\R = 300\;\Omega \\\end{array}$

Ans:

Thus, the value of the unknown resistance is $300\;\Omega$ .