Problem 70 - Master Resource Book in JEE Main Physics

Question

If \(X=A \times B\) and \(\Delta X, \Delta A\) and \(\Delta B\) are maximum absolute errors in \(X, A\) and \(B\) respectively, then the maximum relative error in \(X\) is given by (a) \(\Delta X=\Delta A+\Delta B\) (b) \(\Delta X=\Delta A-\Delta B\) (c) \(\frac{\Delta X}{X}=\frac{\Delta A}{A}-\frac{\Delta B}{B}\) (d) \(\frac{\Delta X}{X}=\frac{\Delta A}{A}+\frac{\Delta B}{B}\)

Step-by-Step Solution

Answer

The maximum relative error in \(X\) is given by (d): \(\frac{\Delta X}{X}=\frac{\Delta A}{A}+\frac{\Delta B}{B}\).

Step 1: Understand the Formulas

The given exercise involves understanding how errors propagate in multiplication. We have \(X = A \times B\), and we need to analyze the maximum relative error in \(X\).

Step 2: Consider Absolute Errors

The absolute errors \(\Delta X\), \(\Delta A\), and \(\Delta B\) represent the maximum potential deviations in each variable, \(X, A,\) and \(B\) respectively.

Step 3: Use Error Propagation for Multiplication

In a multiplication \(X = A \cdot B\), the error propagation principle states that the relative error in the product is approximately the sum of the relative errors in the factors. The formula can be expressed as: \[\frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\] This is consistent with option (d).

Step 4: Conclusion Verification

Verify that this matches with option (d): \(\frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\), confirming our solution is correct.

Key Concepts

Maximum Relative Error

When working with measurements in mathematical operations, it's crucial to understand how errors can affect the final result.
The maximum relative error is an important concept in this context, as it provides insight into the potential deviation of a calculated value from the true value.

• The relative error of a quantity is given by the absolute error divided by the actual value of that quantity.
• For a product like \(X = A \times B\), the relative error shows the percentage by which \(X\) might be off due to errors in \(A\) and \(B\).

To find the maximum relative error of \(X\), we sum up the relative errors from \(A\) and \(B\). This means if \(A\) and \(B\) are both measured values with small uncertainties, these uncertainties add up in their relative forms to impact \(X\). So, the formula is \[\frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\] This equation shows how even small errors in \(A\) and \(B\) can collectively amplify and affect \(X\), providing the maximum expected percentage error in the final result.

Absolute Error

Absolute error refers to the magnitude of error in a measurement, representing how much the measured value deviates from the true value.

• It's essential for understanding error propagation because it provides a concrete numerical value for the degree of uncertainty in a measurement.
• Unlike relative errors, absolute errors include the units of measurement, making them particularly useful when comparing errors in quantities of different magnitudes.

When multiplying quantities, absolute errors play a crucial role. For instance, if we have \(\Delta A\) and \(\Delta B\) as the absolute errors in \(A\) and \(B\), respectively, these values represent the worst-case scenario deviations from their respective actual values.
In multiplication, while the absolute errors themselves aren't directly additive, their relative counterparts are, which contributes to the overall error in the product as indicated in the error propagation principles.

Multiplication Error Analysis

In multiplication error analysis, we determine how errors in individual factors affect the product. Often in scientific and engineering calculations, understanding this helps in estimating the reliability of calculated results.

• Each factor in a multiplication operation contributes to the overall error in the calculated result, \(X\).
• We examine how small errors in the multiplied factors (\(A\) and \(B\)) propagate to affect \(X = A \times B\).

The handy formula for error propagation, \[\frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\], succinctly captures the cumulative effect of these errors. This means that even if \(A\) and \(B\) individually have small inaccuracies, when multiplied, the resulting product \(X\) can still be significantly affected by them.This principle emphasizes the importance of precision in measurements before performing calculations, as the reliability of results can be notably impacted by the extent of initial inaccuracies. This understanding is fundamental in scientific work, ensuring that results are accurate and trustworthy.