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?

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.