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Question
Suppose there are three resistors A, B, and C having resistances r1, r2, 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.
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Solution
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`
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