Question

For the parallel combination of three resistors ,establish the relation.1/Rp=1/R1+1/R2+1/R3 where the symbols have their usual meanings.

Answer 1

Given :

⟶ Three identical resistors of resistanced R$_1$, R$_2$ and R$_3$ are connected in parallel.

To Prove :

➨ 1/R$_p$ = 1/R$_1$ + 1/R$_2$ + 1/R$_3$

Solution :

➠ Let, three resistors are connected in parallel with a battery of voltage V.

✭ Remember :

• Current passing through each resistor remains same in series connection.
• Voltage drop across each resistor remains in parallel connection.

➠ Let, Net current flowing through circuit be I.

◕ Current through R$_1$ = I$_1$

◕ Current through R$_2$ = I$_2$

◕ Current through R$_3$ = I$_3$

➠ As per ohm's law current passing through a conductor is directly proportional to the applied pd.

Mathematically, V = IR

$\longrightarrow\tt\:I=I_1+I_2+I_3\\ \\ \longrightarrow\tt\:\dfrac{V}{R_p}=\dfrac{V}{R_1}+\dfrac{V}{R_2}+\dfrac{V}{R_3}\\ \\ \longrightarrow\tt\:\dfrac{V}{R_p}=V\:{\huge (}\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}{\huge )}\\ \\ \longrightarrow\boxed{\bf{\dfrac{1}{R_p}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}}}$