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Chapter 20: Problem 80

A current-carrying wire of length \(0.15 \mathrm{~m}\) is in a perpendicular magnetic field. The magnetic force on the wire is measured as a function of the current through the wire. The resulting data are given in the table. $$\begin{array}{ll}\hline \text { Current (A) } & \text { Force (N) } \\\\\hline 0.5 & 3.0 \times 10^{-3} \\\0.8 & 9.6 \times 10^{-3} \\\1.4 & 1.7 \times 10^{-2} \\\1.9 & 2.3 \times 10^{-2} \\\\\hline\end{array}$$ Make a plot of the force as a function of the current. Using a "best fit" to the data, determine the magnitude of the magnetic field.

Short Answer

Expert verified
The magnitude of the magnetic field is approximately 0.0812 T.

Step by step solution

01

Understand the Relationship

The magnetic force \( F \) on a current-carrying wire in a magnetic field \( B \) can be expressed by the formula \( F = BIL \), where \( I \) is the current and \( L \) is the length of the wire. Given in the problem is that the wire's length \( L = 0.15 \) m.
02

Rearrange the Formula

To find the magnetic field \( B \), rearrange the equation \( F = BIL \) to solve for \( B \). This gives \( B = \frac{F}{IL} \).
03

Organize Data for Graph

List the given data points of current \( I \) versus force \( F \). These are the pairs: \( (0.5, 3.0 \times 10^{-3}) \), \( (0.8, 9.6 \times 10^{-3}) \), \( (1.4, 1.7 \times 10^{-2}) \), and \( (1.9, 2.3 \times 10^{-2}) \).
04

Plot the Data

Create a plot with the current \( I \) on the x-axis and force \( F \) on the y-axis. Plot each data point: (0.5, 0.003), (0.8, 0.0096), (1.4, 0.017), (1.9, 0.023).
05

Draw the Best Fit Line

Draw a line that best fits all the plotted data points. This line should closely represent the trend shown by the data, minimizing the distance between the line and all points collectively.
06

Calculate the Slope of the Line

The slope of the best fit line gives the ratio \( \frac{\Delta F}{\Delta I} \), which equals \( BL \). Select two points on the line to calculate the slope using \( \frac{F_2 - F_1}{I_2 - I_1} \). Assume points (0.8, 9.6 \times 10^{-3}) and (1.9, 2.3 \times 10^{-2}) for calculations: slope = \( \frac{2.3 \times 10^{-2} - 9.6 \times 10^{-3}}{1.9 - 0.8} = \frac{1.34 \times 10^{-2}}{1.1} = 1.218 \times 10^{-2} \).
07

Compute the Magnetic Field

Using the slope calculated as \( BL \), substitute \( L = 0.15 \) m to find \( B \). \( B = \frac{slope}{L} = \frac{1.218 \times 10^{-2}}{0.15} = 8.12 \times 10^{-2} \). Thus, the magnitude of the magnetic field is approximately \( 0.0812 \) T.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field Calculation
Calculating the magnetic field involves using a well-known formula in physics: the force exerted on a current-carrying wire in a magnetic field is given by \( F = BIL \). Here, \( F \) represents the magnetic force in newtons, \( B \) the magnetic field strength in teslas, \( I \) the current in amperes, and \( L \) the length of the wire in meters. For this exercise, it's crucial to rearrange this formula to solve for \( B \) to determine the magnetic field strength. We achieve this by rearranging it to \( B = \frac{F}{IL} \). This relationship tells us how the magnetic field relates to the measurable quantities of force, current, and length, allowing us to compute \( B \) with given data.
Physics Problem Solving
Solving problems in physics often involves applying known formulas and relationships to specific situations. In our exercise, we need to not only use the formula for magnetic force but also incorporate additional mathematical operations. Physics problem solving often requires:
  • Understanding the underlying principles and relationships, such as \( F = BIL \) here.
  • Reorganizing equations to focus on the desired variable; in this case, rearranging to find \( B \).
  • Systematically collecting and using data, which involves carefully reading the exercise data and understanding its context.
Using these approaches ensures that we can satisfactorily address the main problem through accurate calculation and analysis.
Data Plotting in Physics
Data plotting is an essential part of physics as it helps visualize relationships between variables. For this exercise, we plot force \( F \) as a function of current \( I \). To do this:
  • Label your x-axis with current \( I \) (amperes) and your y-axis with force \( F \) (newtons).
  • Carefully plot each data point provided: \( (0.5, 0.003) \), \( (0.8, 0.0096) \), \( (1.4, 0.017) \), and \( (1.9, 0.023) \).
  • Examine the spread and trend of the data once plotted.
This visual representation is critical as it allows us to detect the pattern or trend of the data visually, which is a precursor to drawing the best fit line.
Slope Determination in Graphs
Determining the slope in a graph relates directly to understanding how one variable changes with another. In the context of our exercise, the slope of the best fit line represents the product \( BL \) (the magnetic field strength times the length of the wire). Calculating the slope requires:
  • Selecting two distinct points that lie on the best fit line for precision. For instance, points \( (0.8, 9.6 \times 10^{-3}) \) and \( (1.9, 2.3 \times 10^{-2}) \) are suitable.
  • Using the slope formula \( \frac{F_2 - F_1}{I_2 - I_1} \) to compute the change in force over the change in current.
  • Interpreting this slope in terms of the physics equation originally used \( F = BIL \), helping us find \( B \) after we substitute the wire length into the calculated slope.
Understanding and accurately determining the slope is key to quantifying these physical relationships.

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Most popular questions from this chapter

Electric fish navigate by responding to changes in the current in seawater. This current is due to a potential difference of around \(3.0 \mathrm{~V}\) generated by the fish and is about 12 mA within a centimeter or so from the fish. Receptor cells in the fish are sensitive to the current. since the current is at some distance from the fish, the sensitivity of these cells suggests that they might be responding to the magnetic field created by the current. To get some estimate of how sensitive the cells are, we can model the current as that of a long, straight wire with the receptor cells \(2.0 \mathrm{~cm}\) away. What is the strength of the magnetic field at the receptor cells?

A closely wound circular coil has a radius of \(6.00 \mathrm{~cm}\) and carries a current of 2.50 A. How many turns must it have if the magnetic field at its center is \(6.39 \times 10^{-4} \mathrm{~T} ?\)

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A closely wound circular coil with a diameter of \(4.00 \mathrm{~cm}\) has 600 turns and carries a current of \(0.500 \mathrm{~A}\). What is the magnetic field at the center of the coil?

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