Problem 69 - Master Resource Book in JEE Main Physics

Question

If \(x=a-b\), then the maximum percentage error in the measurement of \(x\) will be (a) \(\left(\frac{\Delta a+\Delta b}{a-b}\right) \times 100 \%\) (b) \(\left(\frac{\Delta a}{a}-\frac{\Delta b}{b}\right) \times 100 \%_{0}\) (c) \(\left(\frac{\Delta a}{a-a}+\frac{\Delta b}{a-b}\right) \times 100 \%\) (d) \(\left(\frac{\Delta a}{a-a}-\frac{\Delta b}{a-b}\right) \times 100 \%\)

Step-by-Step Solution

Answer

The maximum percentage error in \(x\) is option (a), \(\left(\frac{\Delta a + \Delta b}{a-b}\right) \times 100\%\).

Step 1: Understanding the Variables

The problem gives us two variables, \(a\) and \(b\), and a function of these variables, \(x = a - b\). The percentage error in \(x\) depends on the errors in measuring \(a\) and \(b\), denoted as \(\Delta a\) and \(\Delta b\) respectively. Our goal is to find the maximum percentage error in \(x\).

Step 2: Defining Error Propagation

When dealing with subtraction in error propagation, we add the relative errors to find the maximum error in the result. Therefore, the maximum relative error in \(x = a - b\) is given by \(\frac{\Delta a}{a} + \frac{\Delta b}{b}\). However, since \(x = a-b\), we need to adjust our expression to fit \(x\).

Step 3: Calculating Percentage Error

The maximum absolute error in \(x\) is given by \(\Delta a + \Delta b\). To convert this into a percentage with respect to \(x\), multiply the relative error by 100%. This gives us the expression: \[ \left(\frac{\Delta a + \Delta b}{a-b}\right) \times 100\% \].

Step 4: Selecting the Correct Option

Given the expression we derived, look at the options provided:- Option (a) matches our expression exactly: \(\left( \frac{\Delta a + \Delta b}{a-b} \right) \times 100\%\) Thus, option (a) is correct.

Key Concepts

Percentage Error

Understanding percentage error is essential when dealing with measurements and their uncertainties. In physics, percentage error helps express the accuracy of a measurement. It is the difference between the measured value and the exact value, represented as a percentage of the exact value. This allows scientists and students to understand how close a measurement is to the true value, relative to the size of the measurement itself.To calculate the percentage error, you can use this formula:\[\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{Exact Value}}{\text{Exact Value}} \right) \times 100\%\]A smaller percentage error indicates a more accurate measurement.In the context of the exercise given, the percentage error helps us evaluate how the errors in measuring \(a\) and \(b\) affect the result \(x = a - b\). By using the combined errors \((\Delta a + \Delta b)\) relative to \(x\), we quantify how precise the result is in percentage terms.

Relative Error

Relative error gives a sense of how significant an error is in the context of a measurement. It is a dimensionless number, representing the ratio of the absolute error to the actual measurement. This measure helps to compare errors in different measurements without bias from their magnitudes, essentially describing how much the measured value deviates from the actual value.Here’s how you can compute relative error:\[ \text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}} \]In our example, if our measurements for \(a\) and \(b\) have errors, \(\Delta a\) and \(\Delta b\), we find the relative error in each. For subtraction of variables, as seen in \(x = a - b\), the relative errors add up since errors can propagate. Thus, the combined relative error gives an indication of the reliability of our result \(x\). Without the units, this also aids in comparing the qualities of different measured quantities independently of their sizes.

Error in Measurement

Every measurement has a certain degree of uncertainty, known as the error in measurement. This error could arise from various sources, like instrument precision, user error, or environmental conditions. Understanding and calculating error in measurement is crucial for reporting and verifying results in experiments.Errors are expressed as absolute errors, which means the amount by which a measurement is off from the exact value, or as relative errors, reflecting the significance of the error.- Absolute error is straightforward, indicating how far off a measurement is from the true value. In our exercise, this is represented by \(\Delta a\) and \(\Delta b\).- When you combine these errors in operations, like subtraction for \(x = a - b\), they determine how inaccurate the overall outcome could be. Recognizing and calculating these errors accurately allows us to adjust our measurements and expectations accordingly, ensuring that we account for the inherent uncertainties in physical measurements. It underscores why understanding concepts like percentage error and relative error is fundamental in experimental physics and any scientific exploration.