Question

If \[V = \left( {100 \pm 5} \right)\,V\]and \[I = \left( {10 \pm 0.2} \right)\,A\], find \[\Delta R\].

Answer 1

Hint: Use Ohm’s law to express the resistance. Then you can use the formula for the error in the resistance of the circuit.

Formula Used: \[\dfrac{{\Delta R}}{R} = \pm \left( {\dfrac{{\Delta V}}{V} + \dfrac{{\Delta I}}{I}} \right)\]
Here, \[\Delta R\] is the error in the resistance, R is the resistance, \[\Delta V\] is the error in the voltage and \[\Delta I\] is the error in the current.

Complete step by step answer:
We know that according to Ohm’s law, the voltage in the circuit is proportional to the current.
Therefore, \[V \propto I\].
\[ \Rightarrow V = IR\]

Here, I is the current and R is the resistance in the circuit.

Therefore, we can rearrange the above equation for R.
\[R = \dfrac{V}{I}\]

We have given voltage and current values with maximum and minimum possible error in voltage and current.

The resistance of the circuit is can be calculated using the above equation as follows,
 \[R = \dfrac{{100\,V}}{{10\,A}}\]
\[ \Rightarrow R = 10\,\Omega \]

This is the resistance of the circuit neglecting the error in the resistance.

Now, the error in the resistance of the circuit is,
\[\dfrac{{\Delta R}}{R} = \pm \left( {\dfrac{{\Delta V}}{V} + \dfrac{{\Delta I}}{I}} \right)\]
\[ \Rightarrow \Delta R = \pm \left( {\dfrac{{\Delta V}}{V} + \dfrac{{\Delta I}}{I}} \right)R\]

Substitute 5 V for \[\Delta V\], 100 V for V, 0.2 A for \[\Delta I\], 10 A for I and \[10\,\Omega \] for R in the above equation.
\[\Delta R = \pm \left( {\dfrac{5}{{100}} + \dfrac{{0.2}}{{10}}} \right)\left( {10} \right)\]
\[ \Rightarrow \Delta R = \pm \left( {0.07} \right)\left( {10} \right)\]
\[ \Rightarrow \Delta R = \pm 0.7\,\Omega \]

Therefore, the error in the resistance \[\Delta R\] is \[ \pm 0.7\,\Omega \]. We can now write the value of resistance with maximum and minimum possible error as follows,
\[R = \left( {10 \pm 0.7} \right)\,\Omega \]

Additional information: The percentage in the error of the resistance is expressed as,
\[\dfrac{{\Delta R}}{R} \times 100 = \pm \left( {0.07} \right) \times 100\]
\[ \Rightarrow \dfrac{{\Delta R}}{R} \times 100 = \pm 7\% \]
The error in the resistance is generally given in tolerance which is the percentage of the error in the resistance. For example, if \[R = \left( {25 \pm 5\% } \right)\Omega \] therefore, the error in the measurement of the resistance will be 5%.

Note:While solving these types of questions, the unit of resistance, current and voltage does not affect the final answer as long as change in resistance and original resistance have the same unit. If you want to find the resistance, you can always use Ohm’s law by taking the original values of current and voltage in the circuit.