Skip to main content
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 4
Chapter 31, Problem 4

A rocket zooms past the earth at v=2.0×106 m/s. Scientists on the rocket have created the electric and magnetic fields shown in FIGURE EX31.4. What are the fields measured by an earthbound scientist?

Verified step by step guidance
1
Step 1: Recognize that the problem involves relativistic transformations of electric and magnetic fields. The fields measured by the earthbound scientist will differ due to the relative motion between the rocket and the earth.
Step 2: Use the Lorentz transformation equations for electric and magnetic fields. These equations describe how the fields transform when observed from a reference frame moving at a velocity \( v \) relative to the source of the fields. The transformations are: \( E' = \gamma (E + v \times B) \) and \( B' = \gamma (B - \frac{v \times E}{c^2}) \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \).
Step 3: Identify the components of the electric field \( E \) and magnetic field \( B \) as given in FIGURE EX31.4. These components will be used in the transformation equations. Also, determine the direction of the velocity \( v \) vector relative to the fields.
Step 4: Substitute the given velocity \( v = 2.0 \times 10^6 \, \text{m/s} \) and the components of \( E \) and \( B \) into the Lorentz transformation equations. Compute \( \gamma \) using \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( c \) is the speed of light \( c = 3.0 \times 10^8 \, \text{m/s} \).
Step 5: Simplify the expressions for \( E' \) and \( B' \) to find the transformed electric and magnetic fields as measured by the earthbound scientist. Ensure the cross products \( v \times B \) and \( v \times E \) are calculated correctly based on the directions of the vectors.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8m
Was this helpful?

Key Concepts

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

Relativity of Simultaneity

The relativity of simultaneity is a concept from Einstein's theory of relativity, which states that events that are simultaneous in one frame of reference may not be simultaneous in another. This is crucial for understanding how different observers, such as those on the rocket and those on Earth, perceive the timing and occurrence of events, including the measurement of electric and magnetic fields.
Recommended video:
Guided course
04:27
Intro to Relative Motion (Relative Velocity)

Lorentz Transformation

Lorentz transformations are mathematical equations that relate the space and time coordinates of events as observed in different inertial frames moving relative to each other at constant speeds. These transformations are essential for calculating how electric and magnetic fields change when observed from different frames, particularly when one frame is moving at relativistic speeds, as in the case of the rocket.
Recommended video:
Guided course
13:39
Lorentz Transformations of Velocity

Electromagnetic Field Transformation

Electromagnetic field transformation describes how electric and magnetic fields change when observed from different inertial frames, especially under relativistic conditions. According to Maxwell's equations and the principles of special relativity, the electric field can transform into a magnetic field and vice versa, depending on the relative motion of the observer, which is key to understanding the fields measured by the earthbound scientist.
Recommended video:
Guided course
03:00
Transformers
Related Practice
Textbook Question

What capacitance, in μF, has its potential difference increasing at 1.0×106 V/s when the displacement current in the capacitor is 1.0 A?

1
views
Textbook Question

A 10-cm-diameter parallel-plate capacitor has a 1.0 mm spacing. The electric field between the plates is increasing at the rate 1.0×106 V/m s. What is the magnetic field strength on the axis?

1
views
Textbook Question

Show that the quantity ϵ0(dΦe/dt) has units of current.

1
views
Textbook Question

A rocket cruises past a laboratory at 1.00×106 m/s in the positive x-direction just as a proton is launched with velocity (in the laboratory frame) v⃗=(1.41×106i^+1.41×106j^)\(\vec{v}\) = (1.41 \(\times\) 10^6\,\(\hat{i}\) + 1.41 \(\times\) 10^6\,\(\hat{j}\)) m/s. What are the proton's speed and its angle from the y-axis in the rocket frame?

2
views