Finding magnetic field from magnetic force / semielliptical conductor

In summary, the first conversation discussed a charge Q placed in a uniform magnetic field with a displacement given by r(t) = (at^2+bt, 2bt, a). The velocity v(t) was found to be (2at + 2b, 2b, 0) and the magnetic force exerted on the charge at time t=t' was given by F = Q * sqrt( (2at+2b)^2 + 4b^2 ) * a. The question of finding the additional magnetic field when the magnetic field at time t' has changed was left unanswered. The second conversation discussed a semi-elliptical conductor with a current I in a counterclockwise direction.
  • #1
awelex
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Homework Statement


1) A charge Q is placed in a uniform magnetic field of magnitude a that proceeds in the -z direction. Its displacement is given by r(t) = (at^2+bt, 2bt, a).
a) Find v(t).
c) Find the magnetic force exerted on the charge at time t=t'.
d) Now the magnetic field at time t' has changed; a magnetic force vector F = (0, -ab, 4bt^2) is now exerted by the field on the charge in addition to the original magnetic force vector. If the charge has the same velocity, and y-component of B = 0, what's the additional magnetic field?

2) A semi-elliptical conductor (semimajor axis a, semiminor axis b) is centered at the origin in the xy-plane. The conductor carries a current I a CCW direction.

Write the integral expression for the magnetic field vector due to the semi-elliptical conductor at the origin.


Homework Equations





The Attempt at a Solution


1)
a) v(t) = r'(t) = (2at + 2b, 2b, 0)

c) not sure about this one:
F = Q * v(t') * B * sin(theta); but theta = 90 degrees because B goes in -z direction and v(t) doesn't have a z component. Therefore,
F = Q * sqrt( (2at+2b)^2 + 4b^2 ) * a

d) This is the one I have no idea how to solve. I don't even know how to start. Do I have to use the law of Biot and Savart to find the magnetic field? If so, how? I assume another possibility would be to find the "inverse cross product" (if such a thing exists) of the given force vector - but I was told by our professor we don't have to do that. Any hints?


2) We only have to express the integral, not solve it. This is how far I got:

dB = mu0 / (4*Pi) * (I * dl) / r^2, because dl is perpendicular to the "radius" of the ellipse at any point. But the radius itself varies. How do I express that? I looked up the the equations of an ellipse in polar coordinates, so I assume I could use that. But I'd rather understand what exactly I'm doing than simply plugging in equations from the internet.

Thanks.
 
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  • #2
1.d)biot and savart is for current distributions; why use that?

Go on with the lorentz force law. Vectorially add the magnetic forces and equate to (qv)X(B_net). You'll get 3 equations each along x,y and z directions. Solve them and get your field B.

regarding 2, what is the centre of a semi-elliptical conductor?
 
  • #3
Thanks for your answer, saubhik. For 1d) I assumed that's how I have to solve it, but I was specifically told not to - there must be some property that simplifies the problem.

The center of the semi-elliptical conductor is the the point midway between the two foci.
 

1. How do I determine the direction of the magnetic field from a magnetic force on a semielliptical conductor?

The direction of the magnetic field can be determined using the right-hand rule. If the direction of the current in the conductor is known, point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field.

2. Can I use the same method to find the magnetic field for a circular conductor?

Yes, the same method can be used for a circular conductor. The only difference is that the current is constant along a circular conductor, whereas it varies along a semielliptical conductor.

3. Is the magnetic field the same at all points along the semielliptical conductor?

No, the magnetic field strength varies along the semielliptical conductor. It is strongest at the ends of the major axis and weakest at the ends of the minor axis.

4. How does the strength of the magnetic field change as the current in the conductor increases?

The strength of the magnetic field is directly proportional to the current in the conductor. As the current increases, the magnetic field strength also increases.

5. Can I use this method to find the magnetic field for any shape of conductor?

No, this method is specific to a semielliptical conductor. Different shapes of conductors will have different equations and methods for finding the magnetic field.

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