How to calculate the dipole moment of the spherical shell?

In summary, to calculate the dipole moment of a spherical shell with a surface charge distribution σ = k sinφ, you will need to integrate over the x', y', and z' coordinates. The dipole moment will be directed along the y axis, as the function sinφ is positive in the region 0-π and negative in π-2π. Remember to use spherical coordinates and that da' in spherical coordinates is r2 sinθdθdφ.
  • #1
lrcarr

Homework Statement


A spherical shell of radius R has a surface charge distribution σ = k sinφ.
Calculate the dipole moment of the spherical shell.

Homework Equations


P[/B]' = ∫r' σ(r') da'

The Attempt at a Solution


So I believe my dipole will be directed along the y axis, as the function sinφ is positive in the region 0-π and negative in π-2π. Dipoles point from negative to positive, so the right hand side of my sphere will be positive and left negative.

I also know that da' in spherical co-ordinates is r2 sinθdθdφ.

I also thought that maybe because I know my dipole is in y direction, I can do r' → y → rsinθsinφ. I saw another similar problem where the dipole was orientated in z, and they did r' → z → rcosθ.

However no matter what I do, zero seems to fall out at the end. I feel like my understanding of the whole prime system, or spherical co-ordinates, is maybe confused or I am misunderstanding the affect that the direction of the dipole has on this.

What am I doing wrong?
 
Physics news on Phys.org
  • #2
Remember that dipole moment is a vector. So px = ∫x' σ(r') dA' and so on for y' and z'. There are three integrals to do.
 

FAQ: How to calculate the dipole moment of the spherical shell?

1. How is the dipole moment of a spherical shell defined?

The dipole moment of a spherical shell is defined as the product of the charge on the shell and the distance from the center of the shell to the point where the dipole moment is being calculated.

2. What is the formula for calculating the dipole moment of a spherical shell?

The formula for calculating the dipole moment of a spherical shell is p = Qr, where p is the dipole moment, Q is the charge on the shell, and r is the distance from the center of the shell to the point where the dipole moment is being calculated.

3. Do I need to consider the thickness of the spherical shell when calculating the dipole moment?

No, the thickness of the spherical shell does not affect the calculation of the dipole moment. Only the charge and the distance from the center of the shell are needed to calculate the dipole moment.

4. Can the dipole moment of a spherical shell be negative?

Yes, the dipole moment of a spherical shell can be negative if the charge on the shell is negative. This indicates that the dipole moment vector is pointing in the opposite direction of the electric field.

5. What are the units of the dipole moment of a spherical shell?

The units of the dipole moment of a spherical shell are Coulomb-meters (C-m) or Debye (D), with 1 D = 3.33564 x 10^-30 C-m.

Back
Top