A permanent magnet creates a homogenous magnetic field B~
inside a rectangular region. The magnet is moving with a velocity
~v ⊥ B~ . A particle at rest with mass m and charge Q is on the
path of the magnet. Assume that the magnet has a mass much
larger than the particle and that its speed is such that vm/(BQ)
is much less than the lateral extension of the region with magnetic
field.
1. What will be the velocity of the particle (magnitude and
direction) a long time after the magnetic field has reached
and passed its original position?
2. Determine the initial force on the particle in the reference frame in which it is initially at rest
3. Determine the initial force of the particle in the reference frame in which the magnet is at rest.
1. Well, well, well, looks like we have a magnet and a particle caught in a magnetic field tango! Now, when the magnet moves, it creates a changing magnetic field, which will induce an electric field. This electric field will then exert a force on our particle, causing it to move.
Now, since the magnet has a velocity perpendicular to the magnetic field, that means the magnetic force experienced by the particle will be perpendicular to both the velocity of the magnet and the magnetic field. We call this a "slap-in-the-face" force!
Since we have a particle with mass m and charge Q, we can use the good ol' equation F = Q(v × B), where v is the velocity of the particle and B is the magnetic field. Remember, the force is perpendicular to both!
2. Alright, let's talk about the initial force when the particle is at rest. In the reference frame where the particle is initially at rest, we can remove its velocity element (since it's zero!) and we are left with F = Q(0 × B), which simplifies to F = 0.
Yes, you heard it right, my friend! The initial force on the particle is zero! It's like the particle is saying, "Hey, magnet, nice try, but I'm not budging!"
3. Now, let's hop into the reference frame where the magnet is at rest. In this scenario, we're moving with the magnet, so its velocity is zero in our frame. Now the equation becomes F = Q(v × B), but since v (the velocity of the particle) is zero in our frame, the force will still be zero!
That's right, folks! In the reference frame where the magnet is at rest, the initial force on the particle is still zero. It's like a superhero showdown with a lot of fancy moves, but no actual punches!
So, my friend, to sum it up:
1. The velocity of the particle a long time after the magnetic field has passed will depend on all the forces acting on it. We can't say for certain without more information.
2. The initial force on the particle in the frame where it is initially at rest is zero.
3. The initial force on the particle in the frame where the magnet is at rest is also zero.
I hope I didn't magnetize your brain with all the magnetic mumbo jumbo! If you have any more questions, feel free to ask!
To answer these questions, we need to use the principles of electromagnetism and apply the relevant equations.
1. Calculation of the velocity of the particle after the magnetic field has passed:
The motion of the charged particle in a magnetic field is described by the Lorentz force equation:
F = q * v x B
where F is the force on the particle, q is its charge, v is its velocity, and B is the magnetic field.
Since the magnet is moving perpendicular to the magnetic field, the force on the particle will always be perpendicular to its velocity. As a result, the force does not do any work on the particle, and its kinetic energy remains constant.
Therefore, the magnitude of the velocity of the particle will remain the same as its initial velocity, and its direction will be unchanged.
2. Calculation of the initial force on the particle in the reference frame in which it is initially at rest:
In the reference frame in which the particle is initially at rest, we can analyze the situation as if the magnetic field is moving and the particle is stationary.
In this frame, the magnetic field is moving with a velocity (-v) opposite to the velocity of the magnet. Therefore, we need to take into account the magnetic field created by the moving magnet.
According to the principle of magnetic induction, when a magnetic field changes with time, it induces an electric field. This induced electric field exerts a force on the charged particle, given by:
F = q * E
where E is the induced electric field.
In this case, the induced electric field can be calculated using Faraday's law of induction:
E = -v * B
where v is the relative velocity between the magnet and the particle, and B is the magnetic field.
Substituting the values, the initial force on the particle in the reference frame in which it is initially at rest can be calculated as:
F = q * (-v) * B
3. Calculation of the initial force on the particle in the reference frame in which the magnet is at rest:
In the reference frame in which the magnet is at rest, the particle is moving with a velocity (-v) opposite to the velocity of the magnet.
In this frame, the magnetic force on the particle can be calculated using the standard Lorentz force equation mentioned earlier:
F = q * v x B
Since the velocity of the particle and the magnetic field are in opposite directions, the force exerted on the particle will be in a direction opposite to the velocity of the particle.
Therefore, the initial force on the particle in the reference frame in which the magnet is at rest can be calculated as:
F = q * (-v) x B