Question

Define the relative error in a measurement.
The resistance $R={\displaystyle \frac{V}{I}}$ where $V=(100\pm 5)V$ and $I=(10\pm /0.2)$
Find the relative error in resistance R.

Answer 1

Hint: Relative error is used like a measure of precision. It is basically the ratio of the absolute error of a measurement to the measurement being taken. In other words, this kind of error is relative to the size of the quantity being measured. Relative error has been expressed in percentage and has no units.

Complete step-by-step answer:
First of all let us discuss the relative error. A Relative Error is an experimental error used in the approximation or in the experimental testing and analysis processes. In order to get to know and find the relative error, the absolute error requires to be determined first, because both these errors are dependent on each other. In short relative error is absolute error divided by the magnitude of the true value.
As per given in the question,
$\begin{array}{rl}& V=(100\pm 5)V\\ & I=(10\pm 0.2)A\end{array}$
The resistance is found out using the formula,
$R={\displaystyle \frac{V}{I}}$
Substitute the true value in the equation,
$R={\displaystyle \frac{100}{10}}=10\mathrm{\Omega }$
Now let us consider the relative part, it is given as
${\displaystyle \frac{\mathrm{\Delta }R}{R}}=\pm [{\displaystyle \frac{\mathrm{\Delta }V}{V}}+{\displaystyle \frac{\mathrm{\Delta }I}{I}}]$
Substituting in this equation will give,
${\displaystyle \frac{\mathrm{\Delta }R}{R}}=\pm [{\displaystyle \frac{5}{100}}+{\displaystyle \frac{0.2}{10}}]$
Simplifying the equation will give,
${\displaystyle \frac{\mathrm{\Delta }R}{R}}=\pm {\displaystyle \frac{7}{100}}=\pm 0.07$
Hence we got the relative error in resistance of the material.

Note: It is to be noted that in most of the cases, the unit of measurement of the absolute error will be similar as the unit of measurement taken of the actual value, and hence these units will get cancelled each other. Percent error is always determined as a positive number. In others, it is ok to have either a positive or negative value. Absolute and relative error are two important things. Percent error is used as a part of comprehensive error analysis.