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Hak

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- Homework Statement
- Two charges having equal magnitude and opposite sign are connected by a spring with rest length zero, initially inextended, within a region of space in which there is a uniform magnetic field ##\vec B##. The charges begin to move with velocity ##\vec v## perpendicular to ##\vec B##.

##a##. Describe the system's motion and the forces under examination.

##b##. How many degrees of freedom are there in the system?

##c##. The center of mass is desired to move in uniform rectilinear motion, what initial conditions must be imposed? And what appears in the frame of reference of the center of mass?

- Relevant Equations
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Here is my attempt to answer the questions:

a. The system's motion is a combination of two types of motion: the translational motion of the center of mass and the rotational motion of the charges around the center of mass. The forces acting on the system are the Lorentz force, which is the force exerted by the magnetic field on the moving charges, and the spring force, which is the restoring force that tends to keep the charges at their equilibrium distance. The Lorentz force is given by $$\vec F = q \vec v \times \vec B$$, where $$q$$ is the charge, $$\vec v$$ is the velocity, and $\vec B$ is the magnetic field. The spring force is given by $$\vec F = -k \vec x$$, where $$k$$ is the spring constant and $$\vec x$$ is the displacement from the equilibrium position. The Lorentz force causes the charges to move in circular paths in planes perpendicular to $\vec B$, while the spring force causes them to oscillate along the line joining them.

b. The number of degrees of freedom of a system is the number of independent parameters that can be varied to describe its state. For this system, we can choose three parameters: the position of the center of mass, the angle of rotation of the charges around the center of mass, and the distance between the charges. Therefore, the system has three degrees of freedom.

c. For the center of mass to move in uniform rectilinear motion, it must have a constant velocity that is parallel to $$\vec B$$. This means that the initial velocity of each charge must also be parallel to $$\vec B$$, so that there is no component of velocity perpendicular to $$\vec B$$ that would cause a Lorentz force. In addition, the initial distance between the charges must be equal to their equilibrium distance, so that there is no spring force acting on them. In this case, there will be no net force on the system and it will move with constant velocity along $$\vec B$$. In the frame of reference of the center of mass, the charges will appear to be at rest relative to each other.

Since I am pretty sure the answers are all wrong, could you help me out?

a. The system's motion is a combination of two types of motion: the translational motion of the center of mass and the rotational motion of the charges around the center of mass. The forces acting on the system are the Lorentz force, which is the force exerted by the magnetic field on the moving charges, and the spring force, which is the restoring force that tends to keep the charges at their equilibrium distance. The Lorentz force is given by $$\vec F = q \vec v \times \vec B$$, where $$q$$ is the charge, $$\vec v$$ is the velocity, and $\vec B$ is the magnetic field. The spring force is given by $$\vec F = -k \vec x$$, where $$k$$ is the spring constant and $$\vec x$$ is the displacement from the equilibrium position. The Lorentz force causes the charges to move in circular paths in planes perpendicular to $\vec B$, while the spring force causes them to oscillate along the line joining them.

b. The number of degrees of freedom of a system is the number of independent parameters that can be varied to describe its state. For this system, we can choose three parameters: the position of the center of mass, the angle of rotation of the charges around the center of mass, and the distance between the charges. Therefore, the system has three degrees of freedom.

c. For the center of mass to move in uniform rectilinear motion, it must have a constant velocity that is parallel to $$\vec B$$. This means that the initial velocity of each charge must also be parallel to $$\vec B$$, so that there is no component of velocity perpendicular to $$\vec B$$ that would cause a Lorentz force. In addition, the initial distance between the charges must be equal to their equilibrium distance, so that there is no spring force acting on them. In this case, there will be no net force on the system and it will move with constant velocity along $$\vec B$$. In the frame of reference of the center of mass, the charges will appear to be at rest relative to each other.

Since I am pretty sure the answers are all wrong, could you help me out?

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