Question

Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity \(a\). If the error in the measurement of \(a\) is \(\Delta a(\Delta a / a \ll 1)\), then what is the error \(\Delta r\) in determining \(r ?\) (A) \(\frac{\Delta a}{(1+a)^{2}}\) (B) \(\frac{2 \Delta a}{(1+a)^{2}}\) (C) \(\frac{2 \Delta a}{\left(1-a^{2}\right)}\) (D) \(\frac{2 a \Delta a}{\left(1-a^{2}\right)}\)

Short answer

\(\frac{2 \Delta a}{(1+a)^{2}}\)

Step-by-step solution

Understanding the Problem

We need to find the error \(\Delta r\) in the measurement of the ratio \(r=\frac{(1-a)}{(1+a)}\) caused by the error \(\Delta a\) in the measurement of the quantity \(a\). We'll use the formula for propagation of error for the function \(r(a)\).

Review Propagation of Error for a Function of One Variable

The propagation of error in a function \(f(x)\) when \(x\) is measured with an uncertainty \(\Delta x\) is given by \(\Delta f = \left|\frac{df}{dx}\right|\Delta x\) where \(\frac{df}{dx}\) is the derivative of \(f(x)\) with respect to \(x\).

Differentiate the Given Function \(r(a)\)

The derivative of \(r\) with respect to \(a\) is found by differentiating \(r\) as follows: \[\frac{dr}{da} = \frac{d}{da}\left(\frac{1-a}{1+a}\right) = \frac{-1(1+a) - (1-a)(1)}{(1+a)^{2}} = \frac{-2}{(1+a)^{2}}\]. Since \(\Delta a / a \ll 1\), we can ignore the higher-order terms.

Determine the Error in \(r\)

The error \(\Delta r\) can be calculated using the formula given: \[\Delta r = \left|\frac{dr}{da}\right|\Delta a = \left|\frac{-2}{(1+a)^{2}}\right|\Delta a = \frac{2\Delta a}{(1+a)^{2}}\]. Therefore, the error in determining \(r\) is \(\frac{2 \Delta a}{(1+a)^{2}}\).

Key concepts

Dimensionless Quantity Measurement

Imagine you're given a puzzle with no reference to size or scale. That's a bit like working with dimensionless quantities—numbers without any specific unit of measure. This feature makes dimensionless quantities universal; they're pure numbers that provide a comparison or ratio, lending themselves to broad applications across different fields such as physics, chemistry, and engineering.

When we measure a dimensionless quantity, the focus is on the relative change or comparison between different measurements rather than the actual physical dimensions. In our exercise, the quantity 'a' is such a dimensionless variable, representing perhaps a ratio of lengths, concentrations, or some efficiency factors.

Despite lacking dimensions, errors can still creep into measurements of dimensionless quantities due to instrument precision limitations, observer error, or environmental influences. Careful error propagation is crucial in maintaining accuracy in calculations that involve these tricky, yet widely applicable numbers.

Ratio Error Determination

Errors are the pesky flies at the picnic of precision measurements, and they matter just as much for ratios as for direct measurements. When a measurement involves a ratio, like our exercise's variable 'r', the process of error determination becomes a balancing act of both the numerator and denominator's uncertainties.

In the case of our exercise, determining the error in ratio 'r'—which is derived from the measurement of 'a'—involves understanding that even a small error in 'a' can produce a substantial error in 'r'. This relationship is not always linear, which brings us to the critical role of derivatives in error propagation. The formula for error propagation exploits the derivative to predict how the measurement's uncertainty affects the result.

Knowing the ratio error is vital in fields like experimental physics or engineering, where precision can be the difference between a successful experiment or a bridge's integrity.

Derivative of a Function

The derivative acts like a mathematical microscope, zooming in on the behavior of a function at a particular point. Think of it as a way to detect the slightest change in the function’s value as its input changes. In practical terms, derivatives help us understand rates of change—to see not just how much something is moving, but how fast it's moving.

In the context of our exercise, determining the derivative of function 'r(a)' with respect to the measurement 'a' allows us to quantify how the error in 'a' affects the error in 'r'. This powerful tool is essential for error propagation because it offers a precise rate at which errors in the input measurement propagate through to the output.

Derivatives lay the foundation for differential calculus and are pivotal in fields that require precision and analysis of dynamic systems, such as physics, engineering, economics, and even medicine. Understanding derivatives is like gaining insights into the universe’s changing nature, one tiny slice at a time.