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Consider a particle of mass \(m\) and charge \(q\) moving in a uniform constant magnetic field \(\mathbf{B}\) in the \(z\) direction. (a) Prove that \(\mathbf{B}\) can be written as \(\mathbf{B}=\nabla \times \mathbf{A}\) with \(\mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{r} .\) Prove equivalently that in cylindrical polar coordinates, \(\mathbf{A}=\frac{1}{2} B \rho \hat{\phi}\). (b) Write the Lagrangian (7.103) in cylindrical polar coordinates and find the three corresponding Lagrange equations. (c) Describe in detail those solutions of the Lagrange equations in which \(\rho\) is a constant.

Short Answer

Expert verified
The particle moves in a helical path with constant \( \rho \) and \( z \)-axis velocity.

Step by step solution

01

Understand the Magnetic Vector Potential

The magnetic field \( \mathbf{B} \) is related to the vector potential \( \mathbf{A} \) by the equation \( \mathbf{B} = abla \times \mathbf{A} \). In this problem, the vector potential is given as \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \). We need to verify this expression in cylindrical polar coordinates.
02

Verify the Vector Potential in Cylindrical Coordinates

In Cartesian coordinates \( \mathbf{r} = (x, y, z) \). For a magnetic field aligned in the \( z \)-direction, \( \mathbf{B} = (0, 0, B) \). Calculate \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \): \[ \mathbf{A} = \frac{1}{2} (0, 0, B) \times (x, y, z) = \frac{1}{2}(-By, Bx, 0) \]. In cylindrical coordinates, \( x = \rho \cos(\phi) \), \( y = \rho \sin(\phi) \). Therefore, \( \mathbf{A} = \frac{1}{2} B \rho \hat{\phi} \).
03

Setup the Lagrangian in Cylindrical Coordinates

The Lagrangian under a magnetic field is given by: \[ L = \frac{1}{2} m ( \dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2 ) + q \mathbf{v} \cdot \mathbf{A} \]. Substitute \( \mathbf{A} = \frac{1}{2} B \rho \hat{\phi} \) and \( \mathbf{v} = (\dot{\rho}, \rho \dot{\phi}, \dot{z}) \). This gives \( L = \frac{1}{2}m(\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) + \frac{1}{2}qB \rho^2 \dot{\phi} \).
04

Derive the Lagrange Equations

Using the Euler-Lagrange equation \( \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i}) - \frac{\partial L}{\partial q_i} = 0 \), derive equations for \( \rho \), \( \phi \), and \( z \): 1. For \( \rho \): \[ m \ddot{\rho} - m \rho \dot{\phi}^2 = -qB\rho \dot{\phi} \]2. For \( \phi \): \[ \frac{d}{dt}(m \rho^2 \dot{\phi} + \frac{1}{2} qB \rho^2) = 0 \]3. For \( z \): \[ m \ddot{z} = 0 \] (no magnetic field effect in the z direction).
05

Analyze Solutions with Constant \(\rho\)

If \( \rho \) is constant, then \( \ddot{\rho} = 0 \) and equation 1 becomes \( m\rho \dot{\phi}^2 = qB\rho \dot{\phi} \) leading to \( \dot{\phi} = \frac{qB}{m} \). Equation 2 implies \( \dot{\phi} \) is constant if \( \rho \) is constant. The motion of \( z(t) \) is linear because \( m \ddot{z} = 0 \).
06

Conclusion on Motion Behavior with Constant \( \rho \)

The particle moves in a circular trajectory in the xy-plane with \( \dot{\phi} = \frac{qB}{m} \) and maintains a constant velocity along the z-axis, which describes a helical path.

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Key Concepts

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

Magnetic Vector Potential
The Magnetic Vector Potential, denoted as \( \mathbf{A} \), is a fundamental concept in electromagnetism. It acts behind the scenes to help describe magnetic fields. This potential is related to the magnetic field \( \mathbf{B} \) through the curl operation, expressed as \( \mathbf{B} = abla \times \mathbf{A} \).
The beauty of \( \mathbf{A} \) lies in its ability to simplify the representation of the magnetic field and provide insights into the behavior of charged particles in a magnetic environment. One common form of \( \mathbf{A} \) when the magnetic field is constant and pointed in a specific direction is \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \).
This expression holds true because when you calculate the curl of \( \mathbf{A} \), you retrieve \( \mathbf{B} \), thus verifying the correct representation of the magnetic field with respect to vector potential.
Cylindrical Coordinates
Cylindrical Coordinates provide a convenient way to describe points in 3D space, especially where there is rotational symmetry. The system uses three parameters: radius \( \rho \), angle \( \phi \), and height \( z \).
These coordinates are particularly handy when dealing with problems involving circular or helical motion, as they naturally match the geometry of such paths. The transformation between Cartesian coordinates (\(x, y, z\)) to cylindrical coordinates is straightforward.
  • \( x = \rho \cos(\phi) \)
  • \( y = \rho \sin(\phi) \)
  • \( z = z \)
This aligns with the intuitive understanding of using angled radius (\(\rho\)), rotation (\(\phi\)), and vertical height (\(z\)). Using these coordinates to express a vector potential simplifies the calculations necessary for describing motion in magnetic fields.
Euler-Lagrange Equation
The Euler-Lagrange Equation is a powerful mathematical tool used to derive the equations of motion for a system from its Lagrangian. The Lagrangian \( L \), typically expressed in terms of coordinates and velocities, is given by the difference between kinetic and potential energy.
For a system with coordinates \( q_i \), the Euler-Lagrange equation is: \[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \] This equation allows us to find the paths that make the action integral stationary (minimum/maximum), thereby determining the actual motion of the particle.
In problems dealing with a charged particle in a magnetic field, the Lagrangian incorporates terms from both the particle's velocity and the magnetic vector potential, leading to complicated but insightful dynamical equations, as we see with the discussed step-by-step solution.
Circular Motion
Circular Motion occurs when an object moves along the perimeter of a circle. For a charged particle in a magnetic field, such motion is typical in the plane perpendicular to the field lines.
The force that keeps the particle in circular motion is the magnetic Lorentz force, perpendicular to both the velocity of the particle and the magnetic field. This leads to a constant angular speed or cyclotron frequency \( \omega_c \), given by:
\( \omega_c = \frac{qB}{m} \)
where \( q \) is the charge of the particle, \( B \) is the magnetic field strength, and \( m \) is the mass of the particle.
In a magnetic field, this formula tells us how the particle circulates with a radius dependent on the speed perpendicular to the field, creating a circular path. Understanding this concept helps visualize the paths that charged particles take in a magnetic field.
Helical Path
A Helical Path is a three-dimensional trajectory resembling a spring or coil, often described as a combination of circular motion in one plane and linear motion in another direction.
In the context of a charged particle in a magnetic field, a helical path can arise when the particle has velocity components both parallel and perpendicular to the magnetic field lines.
This occurs because the perpendicular component results in circular motion, while the parallel component allows the particle to move along the direction of the field, forming a helix.
The formula for the pitch or separation between the coils of the helix depends on the component of velocity along the \( z \) direction that is unaffected by the magnetic field. This type of motion is common in plasma physics and helps us understand the behavior of particles trapped in magnetic fields.

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Most popular questions from this chapter

Noether's theorem asserts a connection between invariance principles and conservation laws. In Section 7.8 we saw that translational invariance of the Lagrangian implies conservation of total linear momentum. Here you will prove that rotational invariance of \(\mathcal{L}\) implies conservation of total angular momentum. Suppose that the Lagrangian of an \(N\) -particle system is unchanged by rotations about a certain symmetry axis. (a) Without loss of generality, take this axis to be the \(z\) axis, and show that the Lagrangian is unchanged when all of the particles are simultaneously moved from \(\left(r_{\alpha}, \theta_{\alpha}, \phi_{\alpha}\right)\) to \(\left(r_{\alpha}, \theta_{\alpha}, \phi_{\alpha}+\epsilon\right)\) (same \(\epsilon\) for all particles). Hence show that $$\sum_{\alpha=1}^{N} \frac{\partial \mathcal{L}}{\partial \phi_{\alpha}}=0.$$ (b) Use Lagrange's equations to show that this implies that the total angular momentum \(L_{z}\) about the symmetry axis is constant. In particular, if the Lagrangian is invariant under rotations about all axes, then all components of \(\mathbf{L}\) are conserved.

Write down the Lagrangian for a projectile (subject to no air resistance) in terms of its Cartesian coordinates \((x, y, z),\) with \(z\) measured vertically upward. Find the three Lagrange equations and show that they are exactly what you would expect for the equations of motion.

A mass \(m_{1}\) rests on a frictionless horizontal table. Attached to it is a string which runs horizontally to the edge of the table, where it passes over a frictionless, small pulley and down to where it supports a mass \(m_{2} .\) Use as coordinates \(x\) and \(y\) the distances of \(m_{1}\) and \(m_{2}\) from the pulley. These satisfy the constraint equation \(f(x, y)=x+y=\) const. Write down the two modified Lagrange equations and solve them (together with the constraint equation) for \(\ddot{x}, \ddot{y},\) and the Lagrange multiplier \(\lambda\). Use (7.122) (and the corresponding equation in \(y\) ) to find the tension forces on the two masses. Verify your answers by solving the problem by the elementary Newtonian approach.

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum"- unqualified - is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius \(r=R\) the length of the pendulum. A convenient choice of coordinates is spherical polars, \(r, \theta, \phi,\) with the origin at the point of support and the polar axis pointing straight down. The two variables \(\theta\) and \(\phi\) make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the \(\phi\) equation tells us about the \(z\) component of angular momentum \(\ell_{z^{*}}\) (c) For the special case that \(\phi=\) const, describe what the \(\theta\) equation tells us. (d) Use the \(\phi\) equation to replace \(\dot{\phi}\) by \(\ell_{z}\) in the \(\theta\) equation and discuss the existence of an angle \(\theta_{\mathrm{o}}\) at which \(\theta\) can remain constant. Why is this motion called a conical pendulum? (e) Show that if \(\theta=\theta_{0}+\epsilon,\) with \(\epsilon\) small, then \(\theta\) oscillates about \(\theta_{\mathrm{o}}\) in harmonic motion. Describe the motion of the pendulum's bob.

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a horizontal plane with constant angular velocity \(\omega\). Write down the Lagrangian for a bead threaded on the rod, using \(r\) as your generalized coordinate, where \(r, \phi\) are the polar coordinates of the bead. (Notice that \(\phi\) is not an independent variable since it is fixed by the rotation of the rod to be \(\phi=\omega t\).) Solve Lagrange's equation for \(r(t) .\) What happens if the bead is initially at rest at the origin? If it is released from any point \(r_{\mathrm{o}}>0,\) show that \(r(t)\) eventually grows exponentially. Explain your results in terms of the centrifugal force \(m \omega^{2} r\).

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