Problem 80 - Physics for Scientists and Engineers with Modern Physics

Question

(III) In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field \(B\) of a permanent magnet. The speed of the cart \(v\) and the induced voltage \(V\) were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data: $$ \begin{array}{lrrrrr} \hline \text { Speed, } v(\mathrm{~m} / \mathrm{s}) & 0.367 & 0.379 & 0.465 & 0.623 & 0.630 \\ \text { Induced voltage, } V(\mathrm{~V}) & 0.128 & 0.135 & 0.164 & 0.221 & 0.222 \\ \hline \end{array} $$ (a) Make a graph of the induced voltage, \(V\), vs. the speed, \(v\). Determine a best-fit linear equation for the data. Theoretically, the relationship between \(V\) and \(v\) is given by \(V=B N \ell v\) where \(N\) is the number of turns of the coil, \(B\) is the magnetic field, and \(\ell\) is the average of the inside and outside widths of the coil. In the experiment, \(B=0.126 \mathrm{~T}, N=50,\) and \(\ell=0.0561 \mathrm{~m} .\) (b) Find the \(\%\) error between the slope of the experimental graph and the theoretical value for the slope. \((c)\) For each of the measured speeds \(v\), determine the theoretical value of \(V\) and find the \(\%\) error of each.

Step-by-Step Solution

Answer

The theoretical slope is 0.3539. Calculate the experimental slope from the graph. Compare both to find % error, then calculate % error for each voltage using theoretical values.

Step 1: Plotting the Graph

Using the given data for speed \(v\) and induced voltage \(V\), plot a graph with speed on the x-axis and voltage on the y-axis. This will help visually interpret the relationship between the two variables.

Step 2: Determine the Best-Fit Line

Use linear regression to determine the best-fit line for the plotted points. The equation of this line will be in the form \( V = mv + c \), where \(m\) is the slope and \(c\) is the y-intercept. From calculation or graphing software, find the values of \(m\) and \(c\).

Step 3: Calculating Theoretical Slope

Theoretically, the relation is given by: \[ V = B N \ell v \]Substitute the given values \( B = 0.126 \, \text{T}, \, N = 50, \, \ell = 0.0561 \, \text{m} \) to find:\[ \text{Slope} = B \cdot N \cdot \ell = 0.126 \times 50 \times 0.0561 \approx 0.3539 \]

Step 4: Calculate % Error of the Slope

The % error is calculated using:\[ \%\, \text{error} = \left( \frac{\text{experimental slope} - \text{theoretical slope}}{\text{theoretical slope}} \right) \times 100 \%\] Plug in the experimental and theoretical slope values to derive the error percentage.

Step 5: Determining Theoretical Values of V

For each speed \(v\), calculate the theoretical induced voltage \(V\) using:\[ V = 0.3539 \times v \]Substitute for each speed \(v\) to get corresponding theoretical \(V\) values.

Step 6: Calculate % Error for Each Voltage

For each measured speed \(v\), compute the % error of each measured voltage \(V\) using:\[ \%\, \text{error} = \left( \frac{\text{measured } V - \text{theoretical } V}{\text{theoretical } V} \right) \times 100 \%\] Conduct calculations for all five data points to find each error percentage.

Key Concepts

Induced Voltage

Induced voltage, also known as electromotive force (emf), is generated when a conductor such as a coil moves through a magnetic field. This phenomenon is a hallmark of electromagnetic induction, discovered by Faraday. The induced voltage is fundamentally the change in voltage across a conductor due to the change in magnetic flux. When the coil moves, it "cuts" through the magnetic lines of force, inducing a voltage across the coil.
In the experiment presented, the voltage induced in the coil can be summarized by the equation: \[ V = B N \ell v \]where:

• \(B\) is the magnetic field strength
• \(N\) is the number of turns in the coil
• \(\ell\) is the length of the coil intercepting the magnetic field
• \(v\) is the velocity at which the coil moves

In practical terms, this means the faster the coil moves through the field, the greater the induced voltage, keeping other factors constant.

Magnetic Field

The magnetic field (denoted by \(B\)) is an essential element in the study of electromagnetic induction. It represents a region around a magnet where magnetic forces can be felt. The magnitude of the magnetic field affects how much voltage will be induced in a moving coil positioned within it.
In this experiment, the magnetic field strength was given as 0.126 Tesla (T). This value directly influenced how much voltage was induced as the cart moved the coil through the field. Tesla is a unit of measure for magnetic flux density, thus representing how dense the magnetic lines of force are in a particular area.
Understanding the magnetic field is crucial, as it not only affects the induced voltage but also determines how many turns of the coil (\(N\)), and the length of the coil (\(\ell\)), interact with the field.

Linear Regression

Linear regression is a statistical method applied in this context to find the best linear fit for the relationship between speed (\(v\)) and induced voltage (\(V\)). In essence, it allows us to determine an equation of the line, usually in the form \(V = mv + c\), that most closely represents the trend of the data points.
Why is this important? It helps identify how predictive the relationship is between speed and voltage. In the experimental data, the slope \(m\) was key, as it helped us determine if theoretical calculations matched observed ones. If the data closely follows a linear path, then the real-world application of the theoretical model is strong.
Software or graphing tools typically carry out this process, offering both the slope and the y-intercept as outputs. These values are critical in comparing the experimental results to the theoretical predictions.

Percentage Error

Percentage error is a vital statistic to gauge the accuracy of experimental measurements against theoretical expectations. It reflects the difference between the measured (experimental) value and the theoretical (predicted) value, offering insight into the reliability of the result.To compute percentage error, use the formula:\[\%\, \text{error} = \left( \frac{\text{experimental value} - \text{theoretical value}}{\text{theoretical value}} \right) \times 100\%\]In the context of this experiment, percentage error was calculated for both the slope of the voltage vs. speed graph and for each measured voltage at their corresponding speeds.
This calculation helps identify deviations in the experimental setup or measurement inaccuracies, thus guiding improvements in future experiments. Notably, a low percentage error indicates high accuracy and consistency between theoretical predictions and experimental outcomes.