Combination of Errors | Class 11 Physics Units and Measurements Chapter 2

Understand how errors combine during addition, subtraction, multiplication, division, and powers with formulas, rules, and numerical concepts important for board exams and entrance examinations.

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What is Combination of Errors?

Combination of errors is the method of determining how errors in measured quantities combine when mathematical operations are performed on them.

When physical quantities are added, subtracted, multiplied, divided, or raised to powers, the associated measurement errors also combine according to specific rules.

This concept is very important in Physics because every measurement contains some uncertainty.

1. Errors of a sum or a difference

If two physical quantities A and B have measured values:$$A \pm \Delta A$$

and$$B \pm \Delta B$$

where ΔA and ΔB are their absolute errors.

Let:$$Z = A \pm B$$

Then the maximum absolute error in Z is:

$\Delta Z = \Delta A + \Delta B$

Rule

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Solved Example 1

Two lengths are measured as:$$A = 20 \pm 0.2 \text{ cm}$$$$B = 15 \pm 0.1 \text{ cm}$$

Find the absolute error in:$$Z = A + B$$

Solution

Using the formula:$$\Delta Z = \Delta A + \Delta B$$$$\Delta Z = 0.2 + 0.1$$$$\Delta Z = 0.3 \text{ cm}$$

Final Answer

Absolute error in the result = 0.3 cm

2. Error of a Product

If:$$Z = AB$$

then the maximum relative error is:

$\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

Rule

When two quantities are multiplied, the relative error in the result is the sum of the relative errors in the multipliers.

Solved Example 2

If:$$A = 5 \pm 0.1$$$$B = 10 \pm 0.2$$

Find the relative error in:$$Z = AB$$

Solution

$$\frac{\Delta Z}{Z} = \frac{0.1}{5} + \frac{0.2}{10}$$$$= 0.02 + 0.02$$$$= 0.04$$

Final Answer

Relative error = 0.04

3. Errors in Division

If:$$Z = \frac{A}{B}$$

then the maximum relative error is:

$\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

Rule

When two quantities are divided, the relative error in the result is the sum of the relative errors in the individual quantities.

4. Error in Case of a Quantity Raised to a Power

If:$$Z = A^k$$

then the maximum relative error is:

$\frac{\Delta Z}{Z}=k\frac{\Delta A}{A}$

Rule

The relative error in a physical quantity raised to the power $k$ is $k$ times the relative error in the individual quantity.

Common Mistakes Students Make

Practice Questions

1. Two lengths are measured as:

$$12 \pm 0.2 \text{ cm}$$

and$$18 \pm 0.3 \text{ cm}$$

Find the absolute error in their sum.

2. Find the relative error in:

$$Z = AB$$

if:$$A = 8 \pm 0.2$$

and$$B = 5 \pm 0.1$$

3. A quantity is measured as:

$$A = 4 \pm 0.1$$

Find the relative error in:$$Z = A^3$$