91Ó°ÊÓ

In a laboratory, a rectangular loop of wire surrounds the origin in the \(x z\) -plane, with extent \(H\) in the \(z\) -direction and extent \(L\) in the \(x\) direction (Fig. \(\mathrm{P} 37.55\) ). The loop carries current \(I\) in the counterclockwise direction as viewed from the positive \(y\) -axis. A magnetic field \(\vec{B}=B \hat{k}\) is present. (a) What is the magnetic dipole moment \(\overrightarrow{\boldsymbol{\mu}}\) as seen in the frame \(S\) of the laboratory? (b) What is the torque \(\vec{\tau}\) felt by the current loop? (c) The same loop is viewed from a passing alien spaceship moving with velocity \(\overrightarrow{\boldsymbol{v}}=\boldsymbol{v} \hat{\imath}\) as seen from the laboratory. From the point of view of the aliens, the loop is length contracted in the direction of this motion. What is the magnetic dipole moment \(\overrightarrow{\boldsymbol{\mu}}^{\prime}\) according to the aliens if their coordinate axes are aligned with those of the humans? (d) The torque on the loop is the same when viewed from the laboratory frame and from the spaceship frame. Accordingly, the magnetic field must be framedependent. What is the magnetic field \(\overrightarrow{\boldsymbol{B}}^{\prime}\) in the spaceship frame \(S^{\prime} ?\) (e) If the electromagnetic field is \((\overrightarrow{\boldsymbol{E}}, \overrightarrow{\boldsymbol{B}})=(0, \overrightarrow{\boldsymbol{B}})\) in an inertial frame \(S,\) then in another frame \(S^{\prime}\) moving at velocity \(\overrightarrow{\boldsymbol{v}}\) relative to \(S,\) what is the component of the magnetic field \(\overrightarrow{\boldsymbol{B}}_{\perp}\) perpendicular to \(\overrightarrow{\boldsymbol{v}} ?\)

Step by step solution

01

Calculate Magnetic Dipole Moment (Part a)

Magnetic dipole moment is given by the product of the current, \( I \), and the area, \( A \), of the loop. The loop is rectangular with dimensions \( L \) and \( H \), so it's area is \( A = L \times H \). So, the magnetic dipole moment \( \mu \) in the laboratory frame \( S \) is given by \( \mu = I \times A = I \times L \times H \) in the direction \( \hat{k} \).

02

Calculate Torque on the Current Loop (Part b)

The torque, \( \tau \), on a current loop in a magnetic field is given by the cross product of the magnetic dipole moment and the magnetic field. The magnetic field is in the \( \hat{k} \) direction. So, \( \tau = \mu \times B = 0 \) as the cross product of two parallel vectors is zero.

03

Calculate Magnetic Dipole Moment in Moving Frame (Part c)

According to length contraction in special relativity, the length of the loop in the direction of motion is shortened in the moving frame. As the loop is moving with velocity \( v \) in the \( x \) direction, the length \( L \) is contracted to \( L' = L/\gamma \), where \( \gamma = 1/\sqrt{1 - (v/c)^2} \), \( c \) being the speed of light. The area of the loop in the moving frame is therefore \( A' = L' \times H = L \times H / \gamma \). So the magnetic dipole moment \( \mu' = I \times A' = I \times L \times H / \gamma \) in the \( \hat{k} \) direction.

04

Find the Magnetic Field in the Moving Frame (Part d)

The torque on the loop is the same in both frames. As \( \tau = \mu \times B \), the magnetic field \( B' \) in the moving frame must be such that \( \mu' \times B' = \mu \times B \). As \( \mu' = \gamma \mu \) and \( \tau = 0 \) in both frames, we find that \( B' = B / \gamma \) in the \( \hat{k} \) direction.

05

Find the Magnetic Field Perpendicular to the Velocity (Part e)

The perpendicular component of the magnetic field in the moving frame, \( B'_\perp \), remains the same as in the laboratory frame, \( B_\perp \), as it is perpendicular to the direction of motion. So, \( B'_\perp = B_\perp = B \).

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Key Concepts

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

Torque on Current Loop

Understanding the concept of torque on a current loop is essential in the realms of electromagnetism and physics in general. When a loop of wire carries an electrical current in a magnetic field, it experiences a torque. This torque is a result of the interaction between the magnetic moment of the loop and the external magnetic field, which can cause the loop to rotate unless an external force prevents it.
The equation for torque (\( \vec{\tau} \)) on a current loop is given by the vector cross product:

• \( \vec{\tau} = \vec{\mu} \times \vec{B} \)

Here, \( \vec{\mu} \) is the magnetic dipole moment and \( \vec{B} \) is the magnetic field. The magnetic dipole moment is computed as the product of the current \( I \) and the area of the loop \( A \).
In scenarios where the magnetic field is parallel to the magnetic dipole moment, like in this exercise, the torque is zero because the cross product of two parallel vectors is zero. This plays an important role in applications like electric motors, where controlling torque involves manipulating these parameters.

Length Contraction

Length contraction is a fascinating prediction of Einstein's theory of special relativity. It states that objects moving at a significant fraction of the speed of light will appear shorter in the direction of the motion as observed from a stationary frame of reference. This concept is not only theoretical but can have practical implications in scenarios involving high-speed particles.In the context of the exercise, the spaceship observes the rectangular loop of wire as being contracted. The length in the direction of motion \( L \) contracts, resulting in a contracted length \( L' \). The formula for this contracted length can be derived using the Lorentz factor \( \gamma \):

• \( L' = \frac{L}{\gamma} \)
• \( \gamma = \frac{1}{\sqrt{1 - (v/c)^2}} \)

Here, \( v \) is the velocity of the spaceship and \( c \) is the speed of light. Through this contraction, the perceived area changes, affecting the magnetic dipole moment viewed from the spaceship, highlighting how special relativity affects classical electromagnetic observations.

Special Relativity

Special relativity, formulated by Albert Einstein, revolutionized the way we understand space, time, and how they are interwoven with motion. Central to its propositions are two postulates: the laws of physics are the same in all inertial frames, and the speed of light is the same for all observers, regardless of the motion of the light source or observer.
Among its many implications, special relativity introduces concepts such as time dilation and length contraction. It affects electromagnetic phenomena, leading to transformations of electric and magnetic fields between different frames of reference. This exercise covers how these fields transform when observed from different frames in relative motion,
carefully analyzing how observers on a moving spaceship perceive the magnetic field compared to those in a stationary laboratory. Such transformations ensure that physics equations hold in every inertial frame, showcasing the underlying symmetry and beauty of the laws of physics.

Magnetic Field Transformation

Magnetic field transformation refers to how magnetic fields appear to change when viewed from different reference frames moving relative to each other. This is a key aspect of electromagnetism when applying special relativity. In electromagnetic theory, these transformations ensure that electromagnetic equations appropriately convert between frames as per Lorentz transformations.
In the given exercise, we see that the torque on the loop should remain unchanged whether viewed from the laboratory frame or the spaceship frame. Since torque is the same in both frames, the intensity of the magnetic field must be adjusted to satisfy this requirement. Given the relations involving magnetic dipoles and fields, one can determine the transformed magnetic field \( \vec{B}' \):

• \( \vec{B}' = \frac{\vec{B}}{\gamma} \)

Here, \( \gamma \) is the Lorentz factor, illustrating how magnetic fields contract in terms of intensity but maintain symmetry across different reference frames. Understanding magnetic field transformations is vital for modern physics and technological applications, such as GPS systems and spacecraft navigation.