Question

Suppose there are three resistors A, B, and C having resistances r₁, r_2, and r₃ respectively. If R represents their equivalent resistance, establish the following relation `1/"R" = 1/"r"_1 + 1/"r"_2 + 1/"r"_3`, when joined in parallel.

Answer 1

When joined in parallel, let the main current be i which sub-divides at P into i₁ in r₁, i₂ in r₂ and i₃ in r₃ respectively in given figure.

If potential difference across PQ be V then,

According to Ohm's law:

i₁ = Vr₁   ...(i)

i₂ = Vr₂   ....(ii)

and i₃ = Vr₃    ...(iii)

If R'' be the equivalent resistance of this combination, then

i = `"V"/"R''"`     ....(A)

From (i), (ii) and (iii) on adding, we get   ....(B)

i = i₁ + i₂ + i₃ 

`= "V"/"r"_1 + "V"/"r"_2 + "V"/"r"_3`

`= "V"(1/"r"_1 + 1/"r"_2 + 1/"r"_3)`

Comparing (A) and (B) 

`"V"/"R''" = "V"(1/"r"_1 + 1/"r"_2 + 1/"r"_3)`

or `1/"R''" = 1/"r"_1 + 1/"r"_2 + 1/"r"_3`