Question

The resistance \(\mathrm{R}=\mathrm{V} / \mathrm{I}\) where \(\mathrm{V}=100 \pm 5\) volts and \(\mathrm{I}=10 \pm 0.3\) amperes calculate the percentage error in \(\mathrm{R}\) (a) \(8 \%\) (b) \(10 \%\) (c) \(12 \%\) (d) \(14 \%\)

Short answer

The percentage error in R is \(8\%\), which corresponds to option (a).

Step-by-step solution

Identify the Given Variables

In this problem, we are given the following information: - Voltage (V) = 100 ± 5 volts - Current (I) = 10 ± 0.3 amperes

Calculate the Actual Value of Resistance (R)

We can calculate the actual value of resistance (R) using the formula R = V/I: R = 100 volts / 10 amperes = 10 ohms

Calculate the Percentage Error of Voltage (V) and Current (I)

To find the percentage error of V and I, we use the following formula: \(percentage\: error = \frac{absolute\: error}{measured\: value} \times 100\%\) Percentage error of V: \(\frac{5\: volts}{100\: volts} \times 100\% = 5\%\) Percentage error of I: \(\frac{0.3\: amperes}{10\: amperes} \times 100\% = 3\%\)

Calculate the Percentage Error in Resistance (R)

Since the resistance is calculated using the division of voltage (V) and current (I), we can find the percentage error in R by adding the percentage errors of V and I: Percentage error in R: \(5\% + 3\% = 8\%\) The percentage error in R is 8%, which corresponds to option (a).

Key concepts

Error Propagation

When we're measuring physical quantities such as voltage and current, errors can arise from many sources. Understanding how these errors affect calculations is a key concept known as error propagation. In this context, error propagation helps us determine how uncertainties from different measurements impact our final calculation.

When calculating quantities involving multiple measurements, such as resistance using Ohm's Law (\(R = \frac{V}{I}\),\ where \(V\) is voltage and \(I\) is current), it's important to calculate the percentage error for each measurement. For instance, if the error in the voltage measurement is \(\pm 5\) volts from a \(100\) volt reading, the percentage error is \(5\%\). Similarly, if the error in the current is \(\pm 0.3\) amperes from a \(10\) ampere reading, the percentage error is \(3\%\).

To find the percentage error in the calculated resistance, we add the individual percentage errors of voltage and current. This technique, often called the sum of relative errors, simplifies how propagated errors affect calculated results. In our problem, adding the percentage errors of voltage (\(5\%\)) and current (\(3\%\)) gives us an overall percentage error of \(8\%\) for the resistance. This approach assumes that the errors in measurements are independent and follow a linear model of propagation.

Ohm's Law

Ohm's Law is a fundamental principle in electronics and physics that relates voltage, current, and resistance in electrical circuits. It is succinctly captured by the equation \(V = I \cdot R\), where \(V\) is the voltage across the conductor, \(I\) is the current flowing through the conductor, and \(R\) is the resistance of the conductor.

This relationship implies that if you know two of the three quantities, you can always calculate the third. For example, if we know the voltage and the current flowing through a circuit, as in our exercise where \(V = 100\) volts and \(I = 10\) amperes, we can solve for the resistance. Arranging the formula gives \(R = \frac{V}{I}\), allowing us to find the resistance as \(10\) ohms.

Understanding Ohm's Law doesn't just stop at calculating resistance, voltage, or current; it's also about recognizing how interconnected these quantities are. Changing any of these values significantly affects the others. It's also essential to understand the underlying assumptions, such as linearity and the constant temperature, which hold true in many practical situations.

Resistance Calculation

Resistance calculation is essential for designing and analyzing electrical circuits. Resistance is the property that resists the flow of electric current, impacting how circuits perform under different conditions. It's calculated using the equation \(R = \frac{V}{I}\), derived from Ohm's Law, reflecting the ratio between voltage and current.

For our specific task, with a voltage \(V\) of \(100\) volts and current \(I\) of \(10\) amperes, the resistance is \(10\) ohms. This calculation not only shows how resistance is measured in ohms but also illustrates the practical application of Ohm's Law in determining circuit elements.

However, any measurement can include uncertainty, leading to error in the resistance calculation. By calculating the individual percentage errors of our measures, we gain insight into how accurate our resistance estimation is. It helps us improve circuit designs and ensure components operate within their safe limits. Recognizing and incorporating the possibility of measurement error ensures that the resistance values used in any practical application are reliable and safe for use.