Two resistances are given as R_1=(10 \pm 0.5) \Omega and R_2=(15 \pm 0.5) \Omega . The percentage error in the measurement of equivalent resistance when they are connected in parallel is -

Units and Measurements PYQ — JEE Main 2023 · 6 Apr · Shift 1 · Q56 with Solution

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Correct answer: C

Equivalent resistance in parallel

For two resistors in parallel,

R = \frac{R_1R_2}{R_1+R_2}

Given:

R_1 = (10 \pm 0.5)\,\Omega, \qquad R_2 = (15 \pm 0.5)\,\Omega

First compute the equivalent resistance

R = \frac{10\times 15}{10+15} = \frac{150}{25} = 6\,\Omega

Find fractional error using error propagation

Write

R = \frac{R_1R_2}{R_1+R_2}

Taking logarithm is not convenient directly because of the sum term, so use differential form:

\frac{\Delta R}{R} = \frac{\Delta(R_1R_2)}{R_1R_2} + \frac{\Delta\left(\frac{1}{R_1+R_2}\right)}{\frac{1}{R_1+R_2}}

More simply,

R = \frac{R_1R_2}{R_1+R_2}

so maximum fractional error is

\frac{\Delta R}{R} = \frac{\Delta R_1}{R_1} + \frac{\Delta R_2}{R_2} + \frac{\Delta(R_1+R_2)}{R_1+R_2}

Since for a sum, absolute errors add:

\Delta(R_1+R_2)=\Delta R_1+\Delta R_2 = 0.5+0.5=1

Therefore,

\frac{\Delta R}{R} = \frac{0.5}{10} + \frac{0.5}{15} + \frac{1}{25}

Now calculate:

\frac{0.5}{10} = 0.05

\frac{0.5}{15} \approx 0.0333

\frac{1}{25} = 0.04

So,

\frac{\Delta R}{R} = 0.05 + 0.0333 + 0.04 = 0.1233

Hence percentage error:

0.1233 \times 100 = 12.33\%

This does not match any option, which suggests the above direct rule has been overapplied to the denominator term.

Use the standard shortcut for parallel combination

For two resistors in parallel,

R = \frac{R_1R_2}{R_1+R_2}

Using differential method correctly:

dR = \frac{R_2^2\,dR_1 + R_1^2\,dR_2}{(R_1+R_2)^2}

Thus maximum error,

\Delta R = \frac{R_2^2\,\Delta R_1 + R_1^2\,\Delta R_2}{(R_1+R_2)^2}

Substitute values:

\Delta R = \frac{15^2(0.5) + 10^2(0.5)}{(25)^2}

= \frac{225(0.5) + 100(0.5)}{625}

= \frac{112.5 + 50}{625} = \frac{162.5}{625} = 0.26\,\Omega

Now percentage error in equivalent resistance:

\frac{\Delta R}{R}\times 100 = \frac{0.26}{6}\times 100

= 4.33\%

Option check

A: $2.33$ ❌
B: $5.33$ ❌
C: $4.33$ ✅
D: $6.33$ ❌

Therefore, the correct answer is Option C.

More from Units and Measurements

View written solutionFree

Correct answer: C

Equivalent resistance in parallel

For two resistors in parallel,

R = \frac{R_1R_2}{R_1+R_2}

Given:

R_1 = (10 \pm 0.5)\,\Omega, \qquad R_2 = (15 \pm 0.5)\,\Omega

First compute the equivalent resistance

R = \frac{10\times 15}{10+15} = \frac{150}{25} = 6\,\Omega

Find fractional error using error propagation

Write

R = \frac{R_1R_2}{R_1+R_2}

Taking logarithm is not convenient directly because of the sum term, so use differential form:

\frac{\Delta R}{R} = \frac{\Delta(R_1R_2)}{R_1R_2} + \frac{\Delta\left(\frac{1}{R_1+R_2}\right)}{\frac{1}{R_1+R_2}}

More simply,

R = \frac{R_1R_2}{R_1+R_2}

so maximum fractional error is

\frac{\Delta R}{R} = \frac{\Delta R_1}{R_1} + \frac{\Delta R_2}{R_2} + \frac{\Delta(R_1+R_2)}{R_1+R_2}

Since for a sum, absolute errors add:

\Delta(R_1+R_2)=\Delta R_1+\Delta R_2 = 0.5+0.5=1

Therefore,

\frac{\Delta R}{R} = \frac{0.5}{10} + \frac{0.5}{15} + \frac{1}{25}

Now calculate:

\frac{0.5}{10} = 0.05

\frac{0.5}{15} \approx 0.0333

\frac{1}{25} = 0.04

So,

\frac{\Delta R}{R} = 0.05 + 0.0333 + 0.04 = 0.1233

Hence percentage error:

0.1233 \times 100 = 12.33\%

This does not match any option, which suggests the above direct rule has been overapplied to the denominator term.

Use the standard shortcut for parallel combination

For two resistors in parallel,

R = \frac{R_1R_2}{R_1+R_2}

Using differential method correctly:

dR = \frac{R_2^2\,dR_1 + R_1^2\,dR_2}{(R_1+R_2)^2}

Thus maximum error,

\Delta R = \frac{R_2^2\,\Delta R_1 + R_1^2\,\Delta R_2}{(R_1+R_2)^2}

Substitute values:

\Delta R = \frac{15^2(0.5) + 10^2(0.5)}{(25)^2}

= \frac{225(0.5) + 100(0.5)}{625}

= \frac{112.5 + 50}{625} = \frac{162.5}{625} = 0.26\,\Omega

Now percentage error in equivalent resistance:

\frac{\Delta R}{R}\times 100 = \frac{0.26}{6}\times 100

= 4.33\%

Option check

A: $2.33$ ❌
B: $5.33$ ❌
C: $4.33$ ✅
D: $6.33$ ❌

Therefore, the correct answer is Option C.

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