Problem 50 - Master Resource Book in JEE Main Physics

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 \mathrm{\\{} r e s i s t a n c e ~ o f ~ t h e ~ w i r e ~ i s ~ (a) \(6 \%\) (b) zero (c) \(1 \%\) (d) \(3 \%\)

Step-by-Step Solution

Answer

The error in resistance is \(6\%\).

Step 1: Understand Ohm's Law

According to Ohm's Law, the resistance \( R \) of a wire is given by the equation \( R = \frac{V}{I} \), where \( V \) is the voltage and \( I \) is the current.

Step 2: Analyze Error Propagation

When calculating derived quantities like resistance using measurements with errors, we use error propagation rules. For division, the relative error in \( R \) is the sum of the relative errors in \( V \) and \( I \).

Step 3: Calculate Relative Errors

If the current \( I \) and voltage \( V \) both have a percentage error of \( 3\% \), then the relative error in the resistance \( R \) is the sum of these errors: \( 3\% + 3\% = 6\% \).

Step 4: Conclusion

The error in the value of the resistance is \( 6\% \), as it is the sum of the individual percentage errors of the voltage and current measurements.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in physics that describes the relationship between voltage, current, and resistance in an electrical circuit. It is often stated as \( V = IR \), where \( V \) represents the voltage across the conductor, \( I \) is the current flowing through the conductor, and \( R \) is the resistance of the conductor.
This equation implies that the voltage across a conductor is directly proportional to the current flowing through it, provided the resistance remains constant. This principle is crucial for understanding how electrical circuits function.
In practical terms, if you know the values of any two of these variables, you can calculate the third. For example, if you have a known voltage and current, you can determine the resistance by rearranging the formula to \( R = \frac{V}{I} \). Understanding Ohm's Law is essential for electrical engineering and physics, as it forms the basis for analyzing complex circuit behaviors.

Measuring Electrical Resistance

To measure electrical resistance, one typically measures the current flowing through a conductor and the voltage applied across it. Using the equation from Ohm's Law, \( R = \frac{V}{I} \), we can calculate the resistance of the conductor.
To make these measurements, a voltmeter and an ammeter are usually employed. The voltmeter is connected across the conductor to measure the voltage, while the ammeter is placed in series to measure the current.

• Voltmeter: Measures the potential difference. It must have high resistance to draw minimal current, ensuring accurate voltage readings.
• Ammeter: Measures current flow. It should have low resistance so that it doesn't significantly alter the circuit.

By carefully measuring these values and employing Ohm's Law, one can determine the resistance of the wire or any component within an electrical circuit. Accurate measurements are crucial, as errors in current or voltage directly affect the calculated resistance.

Percentage Error Calculation

Percentage error calculation is key when dealing with measurements in physics, as it helps quantify the accuracy of the observed values compared to the true values. In experiments, percentage errors can arise from limitations in measurement precision.
The percentage error can be calculated using the formula: \( \text{Percentage Error} = \frac{|\text{Measured Value} - \text{True Value}|}{\text{True Value}} \times 100\% \). However, when dealing with derived quantities like resistance, we need to consider error propagation.
For measurements with percentage errors, the error in a calculated resistance is the sum of the individual percentage errors of voltage and current. This is because the resistance is derived from dividing voltage by current, making its relative error the sum of their relative errors.
In the given exercise, both the current and the voltage measurement had a 3% error. Hence, the total error in the resistance is the sum of these errors, resulting in a 6% percentage error. This approach helps students understand not only how to calculate such errors but also the significance of accurate measurement in scientific experiments.