Step 1: Understanding the Concept:
In the realm of experimental physics and engineering, the concept of error analysis is paramount because no measurement is perfectly exact.
An error in measurement represents the discrepancy between the value obtained through an experiment (measured value) and the universally accepted or theoretical value (true value).
To provide a meaningful interpretation of how significant an error is, scientists use "Percentage Error."
This metric normalizes the error relative to the magnitude of the quantity being measured, allowing us to compare the accuracy of measurements across different scales.
For example, an error of 2 units in a measurement of 10 units is much more critical than an error of 2 units in a measurement of 1000 units.
By expressing the error as a percentage, we gain an immediate understanding of the precision and reliability of our experimental setup.
Key Formula or Approach:
The mathematical procedure for calculating percentage error involves three distinct metrics:
1. Absolute Error (\(\Delta x\)): This is the numerical magnitude of the difference between the true value and the measured value.
\[ \text{Absolute Error } (\Delta x) = | \text{True Value} (T) - \text{Measured Value} (M) | \]
2. Relative Error: This represents the absolute error expressed as a fraction of the true value. It is a dimensionless quantity.
\[ \text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}} = \frac{|T - M|}{T} \]
3. Percentage Error: This is the final stage where the relative error is converted into a percentage to make it more intuitive for reporting.
\[ \text{Percentage Error} = \left( \frac{|T - M|}{T} \right) \times 100% \]
Step 2: Detailed Explanation:
Let us systematically apply these steps to the numerical values provided in the question.
The problem identifies the "True Value" (\(T\)) as 100 units and the "Measured Value" (\(M\)) as 98 units.
First, we determine the absolute deviation of the measurement:
\[ \text{Absolute Error} = |100 - 98| = 2 \text{ units} \]
This tells us that the experimenter was "off" by exactly 2 units.
However, to understand the quality of this measurement, we must relate this 2-unit discrepancy to the base value of 100 units.
We calculate the relative error by dividing the absolute error by the true value:
\[ \text{Relative Error} = \frac{2}{100} = 0.02 \]
In decimal form, 0.02 represents the fractional part of the measurement that was incorrect.
To arrive at the percentage error, we multiply this result by 100:
\[ \text{Percentage Error} = 0.02 \times 100% = 2% \]
The simplicity of this specific problem lies in the fact that the true value is 100.
When the denominator of a fraction is 100, the numerator directly represents the percentage.
This is a common strategy used in competitive exams to test whether a student can recognize a shortcut or if they get bogged down in unnecessary calculations.
A percentage error of 2% is generally considered quite low in many introductory laboratory settings, indicating a high degree of accuracy.
Step 3: Final Answer:
The absolute difference is 2 units. Since the true value is 100, the percentage error is calculated as \((2 / 100) \times 100 = 2%\).
Therefore, the correct option is (B).