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Chapter 9: Problem 51

A wire carries a current of \(I=10 \mathrm{~A}\) into the page. With a magnetic field probe, a student measures the \(B\)-field at five points and notes down the results: \((1,2.2) ;(2,0.85) ;(5,0.35) ;(8,0.31) ;(10,0.11)\). The first number in each pair is the distance to the right of the wire in \(\mathrm{cm}\) and the second number is the \(B\)-field in units of \(10^{-4} \mathrm{~T}\). a) Plot the data, then plot the expected curve on the same graph. b) What would you say is the average error of the measurements? c) After you have plotted the data, someone reports that the earth's \(B\)-field at this location is \(0.5\) gauss, pointing to the top of the page. Replot the corrected \(B\)-field. d) What percentage error does the earth's field introduce to the measurements as a function of \(r\) ? e) Would you say the original results were trustworthy?

Short Answer

Expert verified
Based on the step-by-step solution provided, answer the following question: Question: Analyze the given data of the magnetic field at different distances from a wire carrying a current, correct it for the Earth's magnetic field, and discuss the trustworthiness of the original results. Answer: To analyze the data and correct it for the Earth's magnetic field, we first plotted the given data points and the expected curve using the formula for the magnetic field around a long straight wire. We then calculated the average error of the measurements by finding the difference between the measured and expected B-field values for each data point. Next, we corrected the given B-field measurements by subtracting the Earth's B-field from each of the given B-field values. We replotted the corrected B-field data on the same graph. To understand the influence of Earth's magnetic field on the measurements, we calculated the percentage error introduced by the Earth's field for each data point and represented it as a function of distance (r) from the wire. To discuss the trustworthiness of the original results, we evaluated the average error, Earth's B-field influence, and percentage error as a function of r. If the average error, Earth's B-field percentage error, and the percentage error as a function of r are small enough, then the original results can be considered trustworthy. However, if these values are significant, the original results might not be reliable due to the influence of the Earth's magnetic field and other potential sources of error.

Step by step solution

01

Know the formula for the magnetic field around a long straight wire

The formula for the magnetic field around a long straight wire is given by B = (μ₀ * I) / (2 * π * r), where B is the magnetic field, μ₀ is the permeability of free space (\(4π × 10^{-7}T·m/A\)), I is the current, and r is the distance from the wire.
02

Convert the given data points to the same units

Convert the given data points from \(10^{-4}~T\) to Tesla by multiplying the B-field values by \(10^{-4}\). Also, convert the distance from cm to meters by multiplying the distance values by 0.01. The converted data points are: \((0.01, 2.2×10^{-4}) ;(0.02, 0.85×10^{-4}) ;(0.05, 0.35×10^{-4}) ;(0.08, 0.31×10^{-4}) ;(0.10, 0.11×10^{-4})\).
03

Calculate the expected B-field values

Using the formula, calculate the expected B-field values for each distance point and compare it with the given data points.
04

Plot the given data points and the expected curve

Using a graphing tool or software, plot the given data points and the expected curve (B-field vs. distance) on the same graph. #b) Calculate the average error of the measurements#
05

Calculate the difference between the measured and expected B-field values

For each data point, calculate the difference between the measured and expected B-field values.
06

Calculate the average error

Add up the absolute differences found in step 1 and divide by the number of data points to get the average error. #c) Replot the corrected B-field#
07

Convert the Earth's B-field from Gauss to Tesla

Earth's B-field is given as 0.5 Gauss. Convert it to Tesla by multiplying with \(10^{-4}\). So, the Earth's B-field in Tesla is \(0.5×10^{-4}~T\).
08

Correct the given B-field measurements

Subtract the Earth's B-field value (found in step 1) from each of the given B-field values in the data points.
09

Replot the corrected B-field data

Plot the corrected B-field data on the same graph as before. #d) Calculate the percentage error introduced by the Earth's field#
10

Calculate the Earth's B-field percentage error

For each data point, calculate the percentage error introduced by the Earth's B-field by dividing the absolute difference of the Earth's B-field and the measured B-field by the measured B-field.
11

Find the percentage error as a function of r

Represent the percentage error as a function of distance (r) from the wire. #e) Discuss the trustworthiness of the original results#
12

Evaluate the trustworthiness of the original results

Analyze the average error, Earth's B-field influence, and percentage error as a function of r to determine whether the original results are trustworthy or not.

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Key Concepts

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

Magnetic Field Around a Wire
Understanding the magnetic field around a wire is fundamental in physics, especially when studying electromagnetism. When an electric current flows through a wire, it creates a magnetic field that encircles the wire. This field can be described by Ampère's right-hand rule, which states that if the thumb of the right hand points in the direction of the current, the fingers curl in the direction of the magnetic field.

The strength of this magnetic field decreases with distance from the wire and can be mathematically calculated using the formula: \( B = \frac{\mu_0 I}{2 \pi r} \)Here, \(B\) is the magnetic field strength, \(\mu_0\) is the magnetic constant (permeability of free space), \(I\) is the electric current, and \(r\) is the radial distance from the wire. This relationship is instrumental in predicting the behavior of the magnetic field at various distances and is crucial for students to understand, especially when preparing for standardized tests such as the SAT or AP physics exams.In practical applications, factors like Earth's magnetic field can influence measurements. This aspect was considered in our textbook exercise, highlighting the importance of understanding real-world complexities in magnetic field measurements.
SAT Physics Preparation
Students preparing for the SAT subject test in physics need to be well-versed in topics such as electromagnetism, which includes understanding how a magnetic field is generated around a wire carrying current.

In SAT physics, questions are often centered around applying concepts to solve problems. A common task might involve analyzing a set of data points related to magnetic fields, similar to our textbook exercise. Key actions include plotting the B-field's magnitude against the distance from the wire and calculating any discrepancies from the expected behavior. These tasks not only test a student's understanding of the concepts but also their skills in applying those concepts to make accurate predictions and analyze errors.Students should practice exercises involving Earth's magnetic field, the effects of measurement errors, and the drawing of field lines, ensuring they can tackle a variety of questions on the exam. Utilizing graphical analysis, understanding the significance of the permeability of free space, and recognizing the influence of external magnetic fields are all crucial for a strong SAT physics test performance.
AP Physics Review
The Advanced Placement (AP) physics curriculum delves deeply into electromagnetism, where exercises such as the one from our textbook help solidify the students' grasp of the principles.

Learning to quantify the magnetic field around a conducting wire, understanding the inherent errors in measurements, and being able to correct for confounding variables like the Earth's magnetic field are all part of the AP physics review. For instance, the concept of percentage errors as a function of distance (\(r\)) is not just a mathematical exercise but also a lesson in experimental physics. The ability to critique the reliability of data, knowing when and how to apply corrections, and interpreting results within the context of theoretical expectations are all skills evaluated in AP exams.Moreover, the inclusion of Earth's magnetic field in our textbook problem adds a layer of complexity, which helps AP students learn about the nuances of real-life experiments and data analysis — invaluable for those pursuing careers in physical sciences or engineering. Discussions about the trustworthiness of original measurements underscore the importance of critical thinking, a key aspect in the AP physics review process.

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Most popular questions from this chapter

Two identical point charges \(q_1\) and \(q_2\) are at a distance \(r\) apart. If the size of \(q_1\) is doubled and the distance between them tripled, the strength of the electrical force between them (A) goes up by a factor of 3 . (B) goes down by a factor of 3 . (C) goes down by a factor of 9 . (D) goes down by a factor of \(2 / 3\). (E) goes down by a factor of \(2 / 9\)

Until recently, TV sets consisted of what was called a cathode-ray tube. Electrons were boiled off a light bulb filament and passed through plates that deflected the particles; the electron beam went on to strike the TV screen itself and lit up a pixel of phosphor. The beam could be deflected both up and down and sideways (only one direction shown). By deflecting the electron beam and scanning it across the screen a picture was built up. Consider the simplified TV below. An electron beam is shot along the centerline between the two deflector plates. The top plate is held by a power supply (not shown) at a voltage \(V=\) \(+1000\) volts above the bottom plate. The separation between the plates is \(d=4 \mathrm{~cm}\). The length of the plates is \(\ell=8 \mathrm{~cm}\). The distance \(L\) from the end of the deflector plate to the TV screen is \(30 \mathrm{~cm}\). The electrons enter the region between the deflector plates with a velocity \(v=4 \times 10^7 \mathrm{~m} / \mathrm{s}\). a) In which direction is the electric field between the deflector plates? b) In which direction are the electrons deflected? c) What is the force on an electron? (Neglect gravity.) d) What is the acceleration on the electron? e) What is the total deflection of the electron as it hits the TV screen? f) What size magnetic field and in what direction would you need to place between the deflector plates in order to prevent the electrons from being deflected?

Instead of a meter, a device (such as a battery) is attached to the coil that can generate a current. When the device is turned on the bar magnet will (A) remain at rest. (B) be attracted to the coil. (C) be repelled from the coil. (D) shoot through the coil. (E) (B) or (C) depending on which way the current is flowing in the coil

Two negative charges with magnitude \(q\) and \(2 q\) sit at points \((1,0)\) and \((0,1)\) on the \(x\) - and \(y\)-axis, respectively. Which figure best represents the total electric field at the origin? (A) \(\mathrm{A}\) (B) \(\mathrm{B}\) (C) \(\mathrm{C}\) (D) \(\mathrm{D}\) (E) \(\mathrm{E}\)

Two identical plastic balls of mass \(10 \mathrm{gm}\) each are hung by threads with length \(30 \mathrm{~cm}\) from a common point, as shown below. The balls are each charged with the same charge \(q\) and repel each other until they come to rest with a horizontal separation of \(30 \mathrm{~cm}\). a) Sketch the electric field produced by the two balls. b) Draw the force vectors on the right-hand ball. c) What is the charge \(q\) in each ball?
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