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Ch 31: Electromagnetic Fields and Waves
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 31: Electromagnetic Fields and WavesProblem 32d
Chapter 31, Problem 32d

In FIGURE P31.32, a circular loop of radius r travels with speed v along a charged wire having linear charge density λ. The wire is at rest in the laboratory frame, and it passes through the center of the loop. What electric and magnetic fields would an experimenter in the loop's frame calculate at distance r from the current of part c?

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Step 1: Analyze the problem setup. The circular loop is moving with velocity v along a charged wire with linear charge density λ. The wire is stationary in the laboratory frame, and the loop is moving relative to it. This relative motion will result in the generation of both electric and magnetic fields in the loop's frame due to the principles of electromagnetism.
Step 2: Determine the electric field in the loop's frame. In the laboratory frame, the wire has a linear charge density λ, which produces an electric field. The electric field at a distance r from the wire in the laboratory frame is given by the formula: E=λ2πεr. Since the loop is moving parallel to the wire, the electric field in the loop's frame remains the same as in the laboratory frame.
Step 3: Account for the magnetic field due to the motion of the loop. In the loop's frame, the motion of the charged wire relative to the loop creates a current. This current generates a magnetic field. Using the Biot-Savart law, the magnetic field at a distance r from the wire can be calculated as: B=μλv2πr, where μ is the permeability of free space.
Step 4: Combine the electric and magnetic fields in the loop's frame. The electric field is radial and points away from the wire, while the magnetic field forms concentric circles around the wire. The direction of the magnetic field can be determined using the right-hand rule, with the thumb pointing in the direction of the wire's motion relative to the loop.
Step 5: Summarize the results. In the loop's frame, the electric field at distance r is E=λ2πεr, and the magnetic field at distance r is B=μλv2πr. These fields are perpendicular to each other, with the electric field being radial and the magnetic field forming circular loops around the wire.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field from a Charged Wire

A charged wire creates an electric field around it, which can be calculated using Gauss's law. For a straight wire with linear charge density λ, the electric field at a distance r from the wire is given by E = (λ / (2πε₀r)), where ε₀ is the permittivity of free space. This field points radially outward from the wire if the charge is positive.
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Magnetic Field due to Moving Charges

When charges move, they generate a magnetic field. The magnetic field B around a long straight wire carrying current I can be described by Ampère's law, where B = (μ₀I) / (2πr) for a distance r from the wire. In this scenario, the moving loop experiences a magnetic field due to the current in the wire, which is crucial for understanding the forces acting on charges within the loop.
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Lorentz Transformation

The Lorentz transformation is a set of equations in special relativity that relate the space and time coordinates of two observers moving at constant velocity relative to each other. In this context, it helps to analyze how electric and magnetic fields transform when viewed from the frame of the moving loop, allowing for the calculation of the fields experienced by an observer in that frame.
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Related Practice
Textbook Question

What is the force (magnitude and direction) on the proton in FIGURE P31.28? Give the direction as an angle cw or ccw from vertical.

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Textbook Question

A proton is fired with a speed of 1.0×106 m/s through the parallel-plate capacitor shown in FIGURE P31.29. The capacitor's electric field is E =(1.0×105 V/m, down). How does an experimenter in the proton's frame explain that the proton experiences no force as the charged plates fly by?

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Textbook Question

A simple series circuit consists of a 150 Ω resistor, a 25 V battery, a switch, and a 2.5 pF parallel-plate capacitor (initially uncharged) with plates 5.0 mm apart. The switch is closed at t = 0 s. Find the electric flux and the displacement current at t = 0.50 ns.

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Textbook Question

A wire with conductivity σ carries current I. The current is increasing at the rate dI/dt. Evaluate the displacement current for a copper wire in which the current is increasing at 1.0×106 A/s.

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Textbook Question

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Textbook Question

A 10 A current is charging a 1.0-cm-diameter parallel-plate capacitor. What is the magnetic field strength at a point 2.0 mm radially from the center of the capacitor?

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