Imperfectly conducting sphere spinning in a homogeneous B-field

In summary, the conversation discusses a problem involving a sphere with imperfect conductivity and moving charges, where the B-field induced by the charges is disregarded. The equations for the electric and magnetic fields inside the sphere are given, which lead to the determination of the E-field and volume charge density inside. The potential inside is also calculated, taking into account the potential at r=0. It is noted that the total charge inside and on the surface are equal and opposite. The problem then shifts to finding the potential and E-field outside the sphere, for which more information is needed about the dimensions, material properties, and external factors of the sphere and its surroundings.
  • #1
yssi83
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Homework Statement


Link to assignment
The fact that it is imperfectly conducting is supplied so that the charges in the sphere will move with the same angular velocity.
The B-field induced by the moving charges will be disregarded.

Homework Equations


[tex]\vec{F}[/tex]=Q[[tex]\vec{E}[/tex]+[tex]\vec{v}[/tex][tex]\times[/tex][tex]\vec{B}[/tex]]
[tex]\vec{F}[/tex]=0 => [tex]\vec{E}[/tex]=-[tex]\vec{v}[/tex][tex]\times[/tex][tex]\vec{B}[/tex]
The equations above apply inside the sphere. And lead to the E-field inside.

The Attempt at a Solution


I have found the E-field inside, and the volume charge density inside.
E=B[tex]\omega[/tex]x[tex]\hat{x}[/tex]+B[tex]\omega[/tex]y[tex]\hat{y}[/tex]
[tex]\rho[/tex]=-2B[tex]\omega[/tex][tex]\epsilon_{o}[/tex]

This gives the potential inside (have set the potential at r=0 to V_0)
V(r)= V_0 - [tex]\frac{1}{2}[/tex]B[tex]\omega[/tex]r^2 (sin [tex]\theta[/tex])^2

The total charge inside and the total charge on the surface are excactly equal but opposite as one would expect. total charge = 0
Q_inside = -[tex]\frac{8}{3}[/tex][tex]\pi[/tex][tex]\epsilon_{0}[/tex]B[tex]\omega[/tex]R^3

Q_surface = [tex]\frac{8}{3}[/tex][tex]\pi[/tex][tex]\epsilon_{0}[/tex]B[tex]\omega[/tex]R^3

Then the problem is to find the potential and E_field outside the sphere. I can't seem to figure out if the charges on the surface are the only ones contributing.
 
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  • #2


I would like to clarify a few points about the problem before providing a solution. Can you please specify the dimensions and material properties of the sphere and the external region? Also, are there any other external factors that may affect the potential and E-field outside the sphere, such as a surrounding medium or external charges? Lastly, can you provide any additional information about the B-field induced by the moving charges, as it may have an impact on the solution? Once these details are clarified, I will be able to provide a more accurate and comprehensive solution.
 

1. What is an imperfectly conducting sphere?

An imperfectly conducting sphere is a spherical object made of a material that is not a perfect conductor of electricity. This means that it allows some amount of electric current to flow through it, but not as easily as a perfect conductor.

2. How does a spinning imperfectly conducting sphere behave in a homogeneous B-field?

When a spinning imperfectly conducting sphere is placed in a homogeneous magnetic field (B-field), it experiences a force known as the Lorentz force. This force causes the sphere to rotate in a direction perpendicular to both the direction of the B-field and the direction of the sphere's spin.

3. What factors affect the behavior of an imperfectly conducting sphere in a B-field?

The behavior of an imperfectly conducting sphere in a B-field is affected by several factors, including the strength and direction of the B-field, the material and conductivity of the sphere, and the speed and direction of the sphere's spin.

4. How can the behavior of an imperfectly conducting sphere in a B-field be described mathematically?

The behavior of an imperfectly conducting sphere in a B-field can be described using the Lorentz force equation, which takes into account the magnetic field strength and direction, the sphere's velocity and spin, and the electric and magnetic properties of the sphere.

5. What are the practical applications of studying an imperfectly conducting sphere in a B-field?

Studying the behavior of an imperfectly conducting sphere in a B-field has practical applications in fields such as electromagnetism, materials science, and engineering. It can also help us better understand the behavior of planets and other celestial bodies with magnetic fields.

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