Question

If potential \[V = 100 \pm 0.5\] Volt and current \[I = 10 \pm 0.2\] amp are given to us, then what will be the value of resistance
A. \[10 \pm 0.7\] ohm
B. \[5 \pm 2\] ohm
C. \[0.1 \pm 0.2\] ohm
D. none of these

Answer 1

Hint:Using Ohm’s law we find the value of the resistance. The relative errors in measuring the potential and the current will contribute to the measurement of the resistance. So, to find the value of resistance, we find the relative error in the resistance using the relative error formula.

Formula used:
\[V = IR\]
where V is the potential across resistance R when current I is flowing through it.

Complete step by step solution:
The potential is given as \[100 \pm 0.5\]volts
\[V = 100\] Volts and the relative error in finding the potential is 0.5 volts.
 \[\dfrac{{\Delta V}}{V} = 0.5\] Volts
The current is given as \[10 \pm 0.2\] amps
\[I = 10\] Amps and the relative error in finding the current is 0.2 amps.
\[\dfrac{{\Delta I}}{I} = 0.2\] Amps
We need to find the resistance with the relative error.

First we find the value of the resistance using Ohm’s law and then find the relative error in resistance using the relative error formula. The resistance is found using Ohm’s law as,
\[R = \dfrac{V}{I} \\ \]
\[\Rightarrow R = \dfrac{{100}}{{10}}\] Ohms
\[\Rightarrow R = 10\] Ohms
Using relative error, the fractional error in resistance is given as,
\[\dfrac{{\Delta R}}{R} = \dfrac{{\Delta V}}{V} + \dfrac{{\Delta I}}{I} \\ \]
\[\Rightarrow \dfrac{{\Delta R}}{R} = 0.5 + 0.2 \\ \]
\[\therefore \Delta R = 0.7\] Ohms
Hence, the resistance is \[R = 10 \pm 0.7\] ohms.

Therefore, the correct option is A.

Note: The relative error in measurement gives the idea about the deviation of the recorded value with respect to the actual measurement of the quantity. We should be careful while using the relative error formula. We should have to take the absolute value of the error while performing the arithmetic operations.