Magnetic dipole moment of a sphere

In summary, to find the magnetic dipole moment of a spherical shell carrying a uniform surface charge, we can use the equation \vec{m} = \frac{1}{2} \int_{S} \vec{r'} \times \vec{K} (\vec{r'}) da', where K is the surface charge density and v is the velocity of the shell, which is equal to \omega times the radius R. The angle between v and R may change, but this is accounted for by the azimuthal radius, which is the distance from the spin axis to the shell measured parallel to the xy plane. This distance can be used as the r value in the integral, and the
  • #1
stunner5000pt
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Homework Statement


Find the magnetic dipole moment of a spherical shell of radiu R carrying a uniform surface charge sigma, set spinning at angular velocity omega.


Homework Equations


[tex] \vec{m} = \frac{1}{2} \int_{S} \vec{r'} \times \vec{K} (\vec{r'}) da' [/tex]

The Attempt at a Solution


So we got to figure out the surface charge density (since it is a spherical shell)

[tex] K = \sigma v [/tex]

and [tex] v = \omega times R [/tex]
this is where i am doubtful...
the angle between v and R varies from 0 to 2 pi

so this cross product is not unique...
or am i thinking about this the wrong way??

please help!
 
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  • #2
You are right that the angle changes but v will always point in the correct direction. The changing angle ([tex]sin(\theta)[/tex]) accounts for the "azimuthal radius", that is, distance from the spin axis z to the shell measured parallel to the xy plane, that changes with polar angle. That distance is also the r you need to use in your integral.

BTW, polar angle only varies from 0 to [tex]\pi[/tex]
 

1. What is the magnetic dipole moment of a sphere?

The magnetic dipole moment of a sphere is a measure of its ability to produce a magnetic field in response to an external magnetic field. It is defined as the product of the strength of the magnetic field and the area of the loop through which the magnetic field passes.

2. How is the magnetic dipole moment of a sphere calculated?

The magnetic dipole moment of a sphere can be calculated by multiplying the radius of the sphere by the current flowing through it. This is known as the magnetic moment formula and is represented by the equation μ = Iπr², where μ is the magnetic moment, I is the current, and r is the radius of the sphere.

3. What are the units of magnetic dipole moment?

The SI unit for magnetic dipole moment is ampere-meter squared (A⋅m²), but it is commonly expressed in terms of the unit of current (ampere) and distance (meter) as well. In cgs units, the unit of magnetic dipole moment is the erg/gauss.

4. How does the magnetic dipole moment of a sphere affect its magnetic field?

The magnetic dipole moment of a sphere is directly proportional to the strength of the magnetic field it produces. This means that as the magnetic dipole moment increases, so does the strength of the magnetic field. Additionally, the orientation of the magnetic dipole moment relative to the external magnetic field can also affect the strength and direction of the resulting magnetic field.

5. What factors can affect the magnetic dipole moment of a sphere?

The magnetic dipole moment of a sphere is primarily affected by its size and the current flowing through it. Other factors that can influence the magnetic dipole moment include the material properties of the sphere (such as its magnetic susceptibility) and the orientation of the sphere relative to an external magnetic field.

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