Current Electricity — Question 4

Below diagram shows two resistors connected in parallel:

Let I₁ and I₂ be the currents through the resistances R₁ and R₂ respectively, then total current drawn from the battery is

I = I₁ + I₂    [Equation 1]

If potential difference between the two ends A and B is V, then by Ohm's law

current in R₁ is I₁ = $\dfrac{V}{R_1}$

current in R₂ is I₂ = $\dfrac{V}{R_2}$

On adding these,

I₁ + I₂ = $\dfrac{V}{R_1}$ + $\dfrac{V}{R_2}$    [Equation 2]

If the equivalent resistance of the combination between the points A and C is R_p, then total current drawn from the source is

I = $\dfrac{V}{R_p}$    [Equation 3]

Substituting the values of I and I₁ + I₂ from equation 3 and 2 in 1, we get,

$\dfrac{V}{R_p} = V\Big(\dfrac{1}{R_1} + \dfrac{1}{R_2}\Big) \\[0.5em] \Rightarrow \dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$

Thus, in the parallel combination, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances.