Resistance of a given wire is obtained by measuring the current flowing in it and the voltage differ

Q: Why do we add the errors even though the formula has division?

In error analysis, we always calculate the maximum possible error. If the numerator is overestimated and the denominator is underestimated, the errors compound. Hence, we always add fractional errors for both multiplication and division.

Q: What if the formula was $R = V - I$? Would we still add percentage errors?

No! For addition and subtraction, we add the *absolute* errors ($\Delta V + \Delta I$), not the percentage or fractional errors. Percentage errors are only added for multiplication and division.

Q: Does the actual value of the resistance matter for this calculation?

Not at all. The beauty of fractional and percentage errors is that they are independent of the actual measured values. You only need the individual percentage errors to find the final percentage error.

Q: Can the errors cancel each other out in real life?

Yes, they can. If both voltage and current are overestimated by $3\%$, the calculated resistance would actually be perfectly accurate! However, in physics, we must report the *maximum possible uncertainty*, which is why we assume the worst-case scenario and add them.

Q: What happens if a quantity has a power, like $I^2$?

If a quantity is raised to a power, you multiply its fractional error by that power. For example, in $P = I^2 R$, the error in power would be $2 \times (\text{error in } I) + (\text{error in } R)$.

Question

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are $3\%$ each, then error in the value of resistance of the wire is

Options:

(A) $6\%$

(B) zero

(C) $1\%$

(D) $3\%$

Sigma Percentile · Accepted Answer

### The Setup: Measuring the Invisible

Imagine you're an electrical engineer tasked with finding the exact resistance of a mysterious wire. You can't just look at it and know; you have to measure it. So, you set up a circuit. You connect a battery, place an ammeter in series to count the electrons flowing through, and hook up a voltmeter in parallel to measure the electrical pressure pushing them.

Ohm's Law is your trusty guide here: 
$$R = \frac{V}{I}$$

It tells you that resistance is simply the voltage divided by the current. But here's the catch: no measuring instrument is perfect. Your voltmeter might be slightly off, and your ammeter might have a tiny glitch.

### The Master Equation: Propagation of Errors

When you calculate a final value using measurements that have errors, those errors don't just disappear—they combine. In physics, we always prepare for the worst-case scenario. We want to know the *maximum* possible error in our final result.

For quantities that are multiplied or divided, the rule is elegant and simple: their **fractional errors add up**. 
$$\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}$$

Notice that even though we are dividing $V$ by $I$, we **add** their fractional errors. Why? Because if the voltage reading is slightly too high and the current reading is slightly too low, the calculated resistance will be significantly higher than the true value. Adding the errors ensures we capture this maximum deviation.

### Final Calculation: Putting it Together

The problem gives us a gift: the percentage errors are already calculated for us! Both the voltage and the current have a $3\%$ error. 
$$\frac{\Delta V}{V} 	imes 100\% = 3\%$$
$$\frac{\Delta I}{I} 	imes 100\% = 3\%$$

To find the total percentage error in resistance, we just plug these into our master equation:
$$\frac{\Delta R}{R} 	imes 100\% = 3\% + 3\%$$
$$\frac{\Delta R}{R} 	imes 100\% = 6\%$$

And there we have it! The maximum error in our calculated resistance is **$6\%$**. It's a beautiful reminder that in the world of experimental physics, uncertainties always compound, and being aware of them is what makes a good scientist great.

Select Answer:

Visualized Solution

Visual Anchor

Ohm's Law

Error Propagation

Substitution

Calculation

Final Answer

The Way Forward

The Sigma Insight: Errors in Measurement

The Setup

The Master Equation

Final Calculation

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