(A) (i) Derive an expression for the equivalent resistance of three resistors connected in parallel with a battery as shown in figure. (ii) Find the…

As in parallel combination, potential difference applied across each resistor is same. Current flowing through each resistor \(\displaystyle \mathrm{R}_{1}, \mathrm{R}_{2}\) and \(\displaystyle \mathrm{R}_{3}\) is

\[I_{1}=\frac{V}{R_{1}}, I_{2}=\frac{V}{R_{2}}, I_{3}=\frac{V}{R_{3}} \]

Current, \(\displaystyle \mathrm{I}=\mathrm{I}_{1}+\mathrm{I}_{2}+\mathrm{I}_{3}\)

\[\begin{aligned} & \frac{\mathrm{V}}{\mathrm{R}_{\mathrm{p}}}=\frac{\mathrm{V}}{\mathrm{R}_{1}}+\frac{\mathrm{V}}{\mathrm{R}_{2}}+\frac{\mathrm{V}}{\mathrm{R}_{3}} \\ & \frac{1}{\mathrm{R}_{\mathrm{p}}}=\frac{1}{\mathrm{R}_{1}}+\frac{1}{\mathrm{R}_{2}}+\frac{1}{\mathrm{R}_{3}} \end{aligned} \]

As all the resistors are connected in parallel, the potential difference applied across each resistor is $\displaystyle 12$ V

\(\displaystyle \mathrm{V}=\mathrm{IR}\)

\(\displaystyle \mathrm{I}_{1}=\frac{\mathrm{V}}{\mathrm{R}_{1}}=\frac{12}{2}=6 \mathrm{~A}\)

\(\displaystyle \mathrm{I}_{2}=\frac{12}{4}=3 \mathrm{~A}\)

\(\displaystyle \mathrm{I}_{3}=\frac{12}{6}=2 \mathrm{~A}\)

Resistivity is the resistance offered by a wire of unit length and unit area of cross section in an electric circuit.