JEE PYQ: Units & Measurements - Question ID 4238a25f4cbe (JEE Main 2012)

In physics, resistance $R$ of a conductor is defined by Ohm’s law as the ratio of the voltage difference $V$ applied across it to the current $I$ flowing through it: $R = \frac{V}{I}.$ When both $V$ and $I$ are measured experimentally, their uncertainties propagate into the calculated resistance. The key concept here is error propagation in division. If two quantities $x$ and $y$ have percentage errors $\delta x$ and $\delta y$, then the percentage error in their ratio $\frac{x}{y}$ is the sum of the individual percentage errors: $\delta\left(\frac{x}{y}\right) = \delta x + \delta y.$ This rule arises from the logarithmic differentiation of the ratio.

Step 1: Express the resistance formula
We start with Ohm’s law: $R = \frac{V}{I}.$

Step 2: Take natural logarithm on both sides
To handle percentage errors, we take the natural logarithm: $\ln R = \ln V - \ln I.$

Step 3: Differentiate both sides
Differentiating with respect to the measured quantities gives: $\frac{dR}{R} = \frac{dV}{V} - \frac{dI}{I}.$

Step 4: Convert differentials to percentage errors
The differentials $\frac{dV}{V}$ and $\frac{dI}{I}$ represent the relative errors in $V$ and $I$. Converting these to percentage errors (multiplying by 100) yields: $\delta R = \delta V + \delta I,$ where $\delta R$, $\delta V$, and $\delta I$ are the percentage errors in $R$, $V$, and $I$ respectively.

Step 5: Substitute the given percentage errors
The question states that the percentage errors in current and voltage are each 3 %. Hence: $\delta V = 3\%,\quad \delta I = 3\%.$ Substituting into the error propagation formula: $\delta R = 3\% + 3\% = 6\%.$

Step 6: Match with the given options
The calculated percentage error in resistance is 6 %, which corresponds to option A.

A frequent mistake students make is to average the percentage errors or to assume that errors cancel out in division. Some even think the error in resistance is simply the larger of the two errors (3 %). However, the correct rule for division is that percentage errors add. Always remember: $\delta\left(\frac{x}{y}\right) = \delta x + \delta y.$