Question

The resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage error in the measurement of the current and voltage are each 3% respectively, the maximum percentage error in measurement of resistance is: (R = $\dfrac{V}{I}$)
A. 6%
B. 3%
C. 1%
D. zero

Answer 1

Use the given formula for resistance and find the percentage error by writing down the expression for it. Percentage error is the ratio of the error to the absolute value times 100% by definition.

Formula used:
If the formula given is of the type:
$Z = \dfrac{A^p B^q}{C^r}$
then we can write the maximum permissible error in Z as:
$\dfrac{\Delta Z}{Z} = p \dfrac{\Delta A}{A} + q \dfrac{\Delta B}{B} + r \dfrac{\Delta C}{C}$.

Complete answer:
Our expression for resistance is:
$R = \dfrac{V}{I}$ .
Therefore, the maximum permissible error in R can be obtained by making use of the formula for percentage error, so we may write:
$e = \dfrac{\Delta R}{R} \times 100 = \dfrac{\Delta V}{V} \times 100 + \dfrac{\Delta I}{I} \times 100$.
Here, we just made a comparison with the formula for Z and made substitutions at the required places like we kept p = 1 and q = 1 in the formula and kept R in place of Z and V in place of A and I in place of B.
Also, we have multiplied both sides of the expression with 100 so that we may get the error in percentage.
Now, in the question itself, we are given the percentage error in V and I to be 3% for the case of both. Therefore, we simply substitute this in the expression we have. We get,
$e = 3 + 3 = 6$%.

Therefore, the correct answer is option (A).

Additional information:
The formula for fractional error has been derived by taking logarithm on both sides of Z expression and then by taking differentiation. It will give us the required expression for maximum permissible error in Z.

Note:
In the formula used section, the expression given is for fractional error and just to get the terminology correct, we have to multiply both sides by 100% so that we have an expression for percentage error. Also, it might appear to be a little confusing if the denominator variables have to be written in negative powers. It should be remembered that all the errors will get added and p, q, r will be positive.