8.1 Collision?
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.
8.2 Magnetic shield
If there was a way to accelerate a spaceship close to the speed of light to go visit some neighboring star system,
one would still face the problem that every particle the spaceship finds on its way will be a projectile hitting it
with almost the speed of light, causing radiation exposure and a lot of damage on the long run.
Assume that the spaceship can illuminate the space in front of it with a powerful laser that strips away the
electrons from any particle, transforming any matter in a collection of free electrons and of charged positive
particles.
Assume farther that the spaceship has technology that allows it to create a large magnetic field in front of
it. For this problem, assume that the spaceship is 100 m wide, that it moves at a speed of 107 m/s, and that the
magnetic field in front of it is also 100 wide but extends 1000 m in front of it with a (homogenous) magnitude
of 10−4 T.
1. What should be the orientation of the magnetic field at the front of the ship to effectively protect it during
its interstellar travel?
2. Once properly oriented, from which particles will this “magnetic shield” most effectively protect? The
electrons or the other positive particles?
3. Starting from which mass will some particles be able to hit the spaceship? (You will need to think about
what happens, make some drawings, and do the math in order to answer this question.)
4. Can the spaceship be protected from hydrogen atoms found in interstellar space?
8.3 About the meaning of the Poynting vector
Remember the Poynting vector? The magnitude of this vector gives the amount of energy per second that
flows through a unit area (in the direction it gives) because of the presence of a combination of electric and
magnetic field, like in an electro-magnetic wave. The units for the magnitude of the Poynting vector are W/m2
(or J/s/m2).
Now, it has been shown that this interpretation for the Poynting vector can also apply to any combination of
electric and magnetic fields, even static fields that do not create any electromagnetic wave! This is a somewhat
surprising result, and this homework is about playing with it for one particular example.
Two circular parallel plates are being charged (starting from their uncharged state) with equal but opposite
charges at a constant rate. The two plates have a radius R, are centered around the z axis, and are parallel
to each other, with a distance d between them. Because they are being charged at a constant rate, the electric
field between the two plates increases linearly from 0 until after a time t0 it reaches a final value E0.
1. Calculate the magnitude and direction of the Poynting vector that is found between the plates, at a
distance r from the center, while the plates are being charged.
2. The Poynting vector gives the energy flow per unit time at one particular position. Integrate this energy
flow to find the total energy per unit time that flows into the volume between the plates.
3. What is the total energy that flows into this volume from time 0 to t0 (when the field changes from 0 to
E0)?
4. What is the energy density found in the volume between the circular plates once the electric field has
reached E0?
8.1 Collision:
1. The velocity of the particle (magnitude and direction) a long time after the magnetic field has reached and passed its original position can be determined using the Lorentz force equation: F = q(v x B), where F is the force experienced by the particle, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field.
Since the particle is initially at rest, its velocity can be considered as perpendicular to the magnetic field (v ⊥ B). Thus, the force experienced by the particle is given by F = qvB.
To find the velocity, we need to equate the Lorentz force with the centripetal force, as the particle will move in a circular path due to the magnetic field. The centripetal force is given by F = mv^2 / r, where m is the mass of the particle and r is the radius of the circular path.
Equating the two forces, we have qvB = mv^2 / r. Solving for v gives:
v = qBr / m
The magnitude of the velocity can be determined as the magnitude of v = qBr / m. The direction of the velocity will be perpendicular to both the magnetic field and the circular path of the particle.
2. The initial force on the particle in the reference frame in which it is initially at rest can be determined using the Lorentz force equation: F = q(v x B), where F is the force experienced by the particle, q is the charge of the particle, v is the initial velocity of the particle (which is zero since it is at rest), and B is the magnetic field.
Since the particle is initially at rest, its velocity is zero. Therefore, the force experienced by the particle will be zero.
3. The initial force on the particle in the reference frame in which the magnet is at rest can also be determined using the Lorentz force equation: F = q(v x B), where F is the force experienced by the particle, q is the charge of the particle, v is the initial velocity of the particle (which is zero since it is at rest), and B is the magnetic field.
Since the magnet is at rest, its velocity is zero. Therefore, the force experienced by the particle will be zero.
1. To determine the velocity of the particle a long time after the magnetic field has reached and passed its original position, we need to consider the Lorentz force equation:
F = q(vxB)
where F is the force on the particle, q is its charge, v is its velocity, and B is the magnetic field.
Since the particle is at rest initially, its velocity v is 0. Therefore, the force on the particle is also 0. As a result, the particle will remain at rest a long time after the magnetic field has passed its original position.
2. To determine the initial force on the particle in the reference frame in which it is initially at rest, we can again use the Lorentz force equation:
F = q(vxB)
Since the particle is initially at rest, its velocity v is 0. Therefore, the force on the particle will also be 0.
3. To determine the initial force of the particle in the reference frame in which the magnet is at rest, we can use the concept of relative velocities. In the reference frame of the magnet, the particle will appear to have a velocity v in the opposite direction. Therefore, the Lorentz force equation becomes:
F = q(-vxB)
Since the particle's velocity is opposite to the direction of the magnetic field B, the force on the particle will be in the opposite direction of the magnetic field.
Now let's move on to the questions related to the magnetic shield.
1. To effectively protect the spaceship during its interstellar travel, the magnetic field at the front of the ship should be oriented perpendicular to the direction of motion of the spaceship. This means the magnetic field lines should be parallel to the spaceship's trajectory.
2. Once properly oriented, the magnetic shield will most effectively protect the positively charged particles. This is because the magnetic field can deflect charged particles with a non-zero charge in motion. Electrons, which have negative charge, can also be deflected by the magnetic field, but they are more easily influenced by electric fields.
3. In order to determine the starting mass of the particles that will be able to hit the spaceship, we need to consider the conditions under which a particle will not be deflected by the magnetic field. This occurs when the force experienced by the particle due to the magnetic field is equal to or less than the centripetal force acting on it.
To calculate the centripetal force, we can use the equation:
Fc = mv^2 / r
where Fc is the centripetal force, m is the mass of the particle, v is its velocity, and r is the radius of the trajectory.
If the force experienced by the particle due to the magnetic field is equal to the centripetal force, then the particle will just barely miss the spaceship. If the force is less than the centripetal force, then the particle will hit the spaceship.
4. Whether or not the spaceship can be protected from hydrogen atoms found in interstellar space depends on the charge of the hydrogen atoms. If the hydrogen atoms are ionized (charged), then the magnetic field can potentially deflect them and provide some protection. However, if the hydrogen atoms are neutral (uncharged), then the magnetic field will have no effect on their trajectory and they can still hit the spaceship. In this case, additional measures may be needed to protect the spaceship from neutral hydrogen atoms.