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Two resistanc...

Question

Two resistances $r_{1} = (5.0 \pm 0.2) Ω$ and $r_{2} = (10.0 \pm 0.1) Ω$ are connected in parallel. Find the value of equivalent resistance with limits of percentage error.

A

r_{p} = 4 Ω \pm

7%

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B

r_{p} = 3.3 Ω \pm

14%

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C

r_{p} = 3.3 Ω \pm

3.5%

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D

r_{p} = 3.3 Ω \pm

7%

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Solution

The correct option is C $r_{p} = 3.3 Ω \pm$ 7%
Here, $r_{1} = 5 Ω, r_{2} = 10 Ω, Δ r_{1} = 0.2 Ω$ and $Δ r_{2} = 0.1 Ω$

The equivalent resistance, $r_{p} = \frac{r_{1} r_{2}}{r_{1} + r_{2}} = \frac{5 \times 10}{5 + 10} = 3.3 Ω$

$\frac{Δ r_{p}}{r_{p}} = \frac{Δ r_{1}}{r_{1}} + \frac{Δ r_{2}}{r_{2}} + \frac{Δ (r_{1} + r_{2})}{r_{1} + r_{2}} = \frac{0.2}{5} + \frac{0.1}{10} + \frac{0.2 + 0.1}{5 + 10} = 0.07$

Thus, The value of equivalent resistance with limits of % error $= r_{p} \pm Δ r_{p} = 3.3 Ω \pm 7 %$

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