Derive an expression for equivalent resistance when three resistors of resistance $R_{1}, R_{2}$ and $R_{3}$ are connected in parallel.

Consider three resistors $R_{1}, R_{2}$ and $R_{3}$ connected in parallel as shown in figure.

When the current $I$ reaches point ' $a^{\prime}$, it splits into three parts $I_{1}$ going through $R_{1}, I_{2}$ going through $R _{2}$ and $I _{3}$ going through $R _{3} .$ The current will tend to take the path of least resistance.

Since charge must be conserved, therefore, the current I that enters point ' $a^{\prime}$ - must be equal to the current that leaves that point. Therefore, we have

$I = I _{1}+ I _{2}+ I _{3}$ $...(1)$

Since the resistors are connected in parallel, therefore, the potential difference across each must be same, hence by $Ohm's$ law, we have

$I _{1}=\frac{ V }{ R _{1}}, I _{2}=\frac{ V }{ R _{2}}$ and $I _{3}=\frac{ V }{ R _{3}} .$ Substituting in equation $(1),$ we have

$I=\frac{V}{R_{1}}+\frac{V}{R_{2}}+\frac{V}{R_{3}}$ $....(3)$

Hence, from equations $(2)$ and $(3)$, we have

$\frac{ V }{ R _{ P }}=\frac{ V }{ R _{1}}+\frac{ V }{ R _{2}}+\frac{ V }{ R _{3}}$

or $\quad \therefore \quad \frac{1}{ R _{ P }}=\frac{1}{ R _{1}}+\frac{1}{ R _{2}}+\frac{1}{ R _{3}}$ $....(4)$