Problem 206 - Calculus Volume 3

Question

Electrical power \(P\) is given by \(P=\frac{V^{2}}{R},\) where \(V\) is the voltage and \(R\) is the resistance. Approximate the maximum percentage error in calculating power if 120 \(V\) is applied to a \(2000-\Omega\) resistor and the possible percent errors in measuring \(V\) and \(R\) are \(3 \%\) and \(4 \%\), respectively.

Step-by-Step Solution

Answer

The maximum percentage error in power is approximately 10%.

Step 1: Identify the formula for power

The given formula for calculating electrical power is \( P = \frac{V^2}{R} \), where \( V \) is the voltage and \( R \) is the resistance.

Step 2: Determine the expression for relative error

The relative error in a function \( P = \frac{V^2}{R} \) can be expressed as: \( \frac{\Delta P}{P} \approx \left| 2 \cdot \frac{\Delta V}{V} \right| + \left| -1 \cdot \frac{\Delta R}{R} \right| \), where \( \Delta V \) and \( \Delta R \) are the errors in \( V \) and \( R \) respectively.

Step 3: Substitute given percentage errors

Given that the percentage error in measuring \( V \) is \(3\%\), so \( \frac{\Delta V}{V} = 0.03 \). Likewise, the percentage error in measuring \( R \) is \(4\%\), so \( \frac{\Delta R}{R} = 0.04 \). Substitute these into the formula: \( \frac{\Delta P}{P} \approx |2 \cdot 0.03| + |-1 \cdot 0.04| \).

Step 4: Calculate the maximum relative error

Perform the calculations: \( \frac{\Delta P}{P} \approx 0.06 + 0.04 = 0.10 \). This represents the relative error in decimal form.

Step 5: Convert to percentage error

To convert the relative error into a percentage error, multiply by 100: \( 0.10 \times 100 = 10\% \). Therefore, the maximum percentage error in calculating power is approximately 10\%.

Key Concepts

Electrical Power Calculation

Calculating electrical power involves understanding how voltage and resistance determine the power output. Electrical power is generally represented by the formula: \( P = \frac{V^2}{R} \), where:

• \( P \) is the power (measured in watts)
• \( V \) is the voltage (measured in volts)
• \( R \) is the resistance (measured in ohms)

This formula shows that power relates directly to the square of the voltage and inversely to resistance. This means if the voltage increases, power increases significantly. Conversely, if resistance increases, power decreases. Calculating now requires precise measurements of both voltage and resistance, as any errors can directly affect the power calculation.

Error Analysis

Error analysis is crucial in accurately determining the outcomes of any calculation, particularly in fields like electronics. The process involves evaluating the extent of errors in measured quantities and understanding how these errors propagate through the calculations to impact the final result. Specifically, for the formula \( P = \frac{V^2}{R} \), we assess how measurement errors in voltage \( V \) and resistance \( R \) affect the calculation of power \( P \). The concept of error propagation shows that errors in variables influence the error in the final computed result. Therefore, identifying and minimizing these 'input' errors can significantly enhance the accuracy of the outcome.

Relative Error

Relative error provides a scale of how significant the error is compared to the actual measurement. This is vital when dealing with variations in measurements, such as voltage and resistance. Relative error is calculated as:\[ \text{Relative Error} = \frac{\Delta X}{X} \]where \( \Delta X \) is the absolute error, and \( X \) is the measured value. In the context of electrical power \( P = \frac{V^2}{R} \), relative error is expressed as:

• \( \frac{\Delta P}{P} \approx \left| 2 \cdot \frac{\Delta V}{V} \right| + \left| -1 \cdot \frac{\Delta R}{R} \right| \)

This shows that relative error considers how errors in both voltage and resistance contribute to the error in calculated power. Understanding and managing relative error allows for better control of accuracy in scientific and engineering calculations.

Percentage Error in Measurements

Percentage error in measurements is an approach to express relative error as a percentage, making it easily understandable and interpretable in practical terms. To convert relative error to percentage error, simply multiply by 100:\[ \text{Percentage Error} = \left( \frac{\Delta X}{X} \right) \times 100 \% \]In the example of calculating power, the analysis led to a maximum relative error of 0.10. Thus, the percentage error becomes:

• \( 0.10 \times 100 = 10\% \)

This means if the error in measuring voltage and resistance is within the given limits of 3% and 4% respectively, then the calculated power could deviate from its true value by up to 10%. Understanding and using percentage error helps communicate the possible deviation in a more intuitive manner, critical for effective decision-making based on measurements.