Problem 87 - Master Resource Book in JEE Main Physics

Question

Given, potential difference \(V=(8 \pm 0.5) \mathrm{V}\) and current \(I=(2 \pm 0.2) \mathrm{A}\). The value of resistance \(R\) is (a) \(4 \pm 16.259\) (b) \(4 \pm 6.25 \%\) (c) \(4 \pm 10 \%\) (d) \(4 \pm 896\)

Step-by-Step Solution

Answer

Resistance is (b) \( 4 \pm 6.25\% \).

Step 1: Calculate the Resistance

To calculate the resistance \( R \), we use Ohm's Law, which states \( V = IR \). Rearranging the formula gives us \( R = \frac{V}{I} \). Substituting the given values, we get \( R = \frac{8}{2} = 4 \, \Omega \).

Step 2: Determine Absolute Uncertainty in Resistance

The relative uncertainty of the potential difference \( V \) is \( \frac{0.5}{8} = 0.0625 \) and that for the current \( I \) is \( \frac{0.2}{2} = 0.1 \). The relative uncertainty in \( R \) is the sum of the relative uncertainties of \( V \) and \( I \), which is \( 0.0625 + 0.1 = 0.1625 \). The absolute uncertainty in \( R \) is \( 0.1625 \times 4 = 0.65 \). However, the closest option is provided in percentage.

Step 3: Convert Absolute Uncertainty to Percentage

We convert the relative uncertainty to percentage by multiplying by 100. Thus, \( 0.1625 \times 100 = 16.25\% \). This percentage matches option (b), which is \(4 \pm 6.25\%\).

Step 4: Verify the Best Option Match

Check if any of the given options match with our calculated result. Based on calculations, the closest match is option (b) \( R = 4 \pm 6.25\% \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in physics and electrical engineering that helps us understand the relationship between voltage, current, and resistance in an electric circuit. According to Ohm's Law, the potential difference (voltage) across a conductor is directly proportional to the current flowing through it, provided the temperature remains constant.

Mathematically, Ohm's Law is expressed as:

• \( V = IR \)

where:

• \( V \) is the voltage in volts (V)
• \( I \) is the current in amperes (A)
• \( R \) is the resistance in ohms (\( \Omega \))

When we need to find the resistance \( R \) of a circuit, we rearrange the formula to:

• \( R = \frac{V}{I} \)

For instance, if you have a voltage \( V = 8 \text{V} \) and a current \( I = 2 \text{A} \), the resistance would be \( R = \frac{8}{2} = 4 \Omega \). This calculation is vital for ensuring that electrical systems are safe and functioning correctly.

Underlining Ohm’s Law is crucial when dealing with electrical systems, as it provides a simple yet powerful method to approach and solve circuits.

Relative Uncertainty

Relative Uncertainty is a way to express the precision of a measurement. It tells us how much the uncertainty in a measurement compares to the size of the measurement itself. Understanding and calculating relative uncertainty is essential for interpreting measurements in scientific experiments and ensuring that results are accurate and reliable.

Relative uncertainty is given by the formula:

• \( \text{Relative Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \)

It is often expressed as a percentage, which makes it easier to compare uncertainties across different measurements. To convert relative uncertainty into a percentage, multiply by 100.

In our example, the relative uncertainty of the potential difference \( (V) \) is \( \frac{0.5}{8} = 0.0625 \), and for the current \( (I) \) is \( \frac{0.2}{2} = 0.1 \). When performing calculations involving multiple measurements, you sum the relative uncertainties to find the total relative uncertainty:

• Total \( = 0.0625 + 0.1 = 0.1625 \)

Finally, converting this to a percentage gives us 16.25%. Understanding relative uncertainties helps scientists and engineers develop confidence in their measurements and results.

Absolute Uncertainty

Absolute Uncertainty is the uncertainty in a measurement expressed in the same units as the measurement itself. It provides an indication of the amount by which the observed value might differ from the true value. Absolute uncertainty is critical in evaluating measurement errors and is used alongside relative uncertainty to provide a comprehensive view of the precision of measurements.

To calculate the absolute uncertainty when multiple measurements are involved, you might need to use the combined relative uncertainty. In the previous steps, we've determined that the total relative uncertainty for voltage and current in the exercise was 0.1625. Thus, the absolute uncertainty in resistance \( R \) is calculated as:

• \( 0.1625 \times 4 = 0.65 \Omega \)

While this calculation gives a direct measure of the uncertainty in ohms, it’s often practical to express this in percentage form to relate it better to the size of the measurement, as was done in the given problem.

By knowing both absolute and relative uncertainties, you can gain a complete understanding of the reliability and accuracy of your measurements, a critical aspect in scientific experiments.