Current density of rotating spherical shell

In summary, the current density of a spherical shell with radius a and surface charge density sigma, rotating with angular velocity omega, will have a phi component given by J_phi = (sigma)*a*Sin(theta)*(omega)*d(r'-a), where the delta function restricts the current density to the surface of the sphere. There is no need to restrict theta to a specific range.
  • #1
icelevistus
17
0
Express the current density of a spherical shell of radius a, rotating with angular velocity omega, with surface charge density sigma



Delta function will be denoted d(x). Spherical coordinates will be used



I have concluded that for a given chunk (if we restrict to the 0<theta<pi/2 domain), the velocity will be given by v=a*Sin(theta)*(omega). It is clear that the current density will only have a phi component. I have concluded:

J_phi = (sigma)*a*Sin(theta)*(omega)*d(r'-a)

Where the delta function is used to restrict the current density to the sphere's surface.

Can anyone confirm that this reasoning is correct?






 
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  • #2
It looks good, except I see no reason for you to restrict theta to [itex]0<\theta<\frac{\pi}{2}[/itex]...what is wrong with your expression for the lower half of the sphere?
 
  • #3
gabbagabbahey said:
It looks good, except I see no reason for you to restrict theta to [itex]0<\theta<\frac{\pi}{2}[/itex]...what is wrong with your expression for the lower half of the sphere?

you're right, as I was writing it I thought I was using cosine and it would introduce a sign error. Whole shell it is!
 

1. What is the definition of current density of a rotating spherical shell?

The current density of a rotating spherical shell refers to the amount of electric current per unit area that is flowing through the surface of the sphere. It is typically denoted by the symbol J and is measured in units of Amperes per square meter (A/m^2).

2. How does the rotation of the spherical shell affect the current density?

The rotation of the spherical shell does not have a direct impact on the current density. However, it can affect the direction of the current flow due to the Coriolis effect, which is a result of the rotation of the Earth.

3. What is the formula for calculating the current density of a rotating spherical shell?

The formula for calculating the current density J of a rotating spherical shell is given by J = I/(4πr^2), where I is the total current flowing through the shell and r is the radius of the sphere.

4. How does the current density of a rotating spherical shell change with distance from the center?

The current density of a rotating spherical shell decreases as the distance from the center of the sphere increases. This is because the area of the spherical shell increases with distance, causing the current to spread out over a larger surface area.

5. What are some real-world applications of the current density of a rotating spherical shell?

The current density of a rotating spherical shell has many applications in electromagnetism, such as in the design of motors and generators. It is also used in geophysics to study the Earth's magnetic field and in astrophysics to understand the dynamics of stars and planetary magnetic fields.

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