Animated Solution for Physics - Physics and Measurement: The resistance $R=\frac{V}{I}$, where $V=(50\pm 2)\text{ V}$ and $I=(20\pm 0.2)\text{ A}$. The percentage error in $R$ is $x$ %. The value of $x$ to the nearest integer is……… .

Imagine you are standing in a physics laboratory, surrounded by wires, batteries, and measuring instruments. You are tasked with finding the resistance of an unknown wire. You set up a simple circuit, connecting a voltmeter in parallel and an ammeter in series.

The voltmeter gives you a reading of $50\text{ V}$, but no instrument is perfect. It has an uncertainty of $\pm 2\text{ V}$. Similarly, the ammeter reads $20\text{ A}$, with an uncertainty of $\pm 0.2\text{ A}$. Your goal is to determine the total percentage error in the calculated resistance.

To find the resistance, we rely on Ohm's law, which states that resistance is the ratio of voltage to current:

But how do we handle the uncertainties? When physical quantities are multiplied or divided, their maximum relative errors are added together. This is a fundamental principle of error propagation. We never subtract errors because we always want to find the maximum possible uncertainty in our final result.

The formula for the percentage error in resistance is:

Let's substitute our given values into the error equation. For voltage, the absolute error $\Delta V$ is $2$, and the base value $V$ is $50$. For current, the absolute error $\Delta I$ is $0.2$, and the base value $I$ is $20$.

Now, let's compute the error contributed by each measurement individually. First, the voltage measurement:

Next, we calculate the percentage error from the current measurement:

Finally, we combine these individual errors to find the total maximum percentage error in the resistance.

Therefore, the value of $x$ is $5$. This is a classic, straightforward question that beautifully illustrates how uncertainties compound when we perform mathematical operations on measured quantities.