Finding Dipole Moment: Solve Integral & Calculate \vec p

In summary, we are given a sphere with a surface charge density of k*cos(theta). We are asked to find the dipole moment, represented as vector p. To do so, we need to use the equation p = integral of r' * charge density, where r' is equal to the unit vector in the direction of the dipole moment. The book assumes the direction of p to be in the z direction, making r' equal to the unit vector in the z direction. However, if the direction of p is unknown, we can resolve vector r into x, y, and z components and calculate these components separately through integrals. This will show that the x and y components cancel, giving us the correct answer.
  • #1
yungman
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242

Homework Statement



Given a sphere radius R with surface charge density [itex]\rho_s=k\;cos\theta[/itex]. Find the dipole moment [itex]\;\vec p[/itex].

Homework Equations



[tex]\vec p=\int \vec r'\rho_s \;d a = \int_0^{2\pi}\int_0^{\pi}\vec r' k\;cos\theta\; R^2d\theta\;d\phi [/tex]

The Attempt at a Solution



To me, [itex]\vec r' = \hat R R[/itex] in spherical coordinates. But the book claimed from the charge density distribution, [itex]\vec p = \hat z p[/itex] which make sense so the book assumed [itex]\;\vec r'=\hat z z = \hat z R\;cos\theta[/itex]. This all make sense.

My real question is what if I don't know the direction of the [itex]\vec p[/itex] by looking at the charge distribution, how am I going to do the integration and find [itex]\vec p[/itex]? If I just use [itex]\vec r' = \hat R R[/itex], the answer won't be correct. Please help.
 
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  • #2
Resolve the vector r into x, y, and z components, expressed in polar coordinates and calculate these components by separate integrals. You will see that both the x and y components cancel.

ehild
 
  • #3
ehild said:
Resolve the vector r into x, y, and z components, expressed in polar coordinates and calculate these components by separate integrals. You will see that both the x and y components cancel.

ehild

I got it, thanks for your help.
 

Related to Finding Dipole Moment: Solve Integral & Calculate \vec p

1. What is a dipole moment and why is it important?

A dipole moment is a measure of the separation of positive and negative charges within a molecule. It is important because it provides information about the polarity and overall charge distribution of a molecule, which can impact its physical and chemical properties.

2. How do you find the dipole moment of a molecule?

The dipole moment can be found by solving the integral of the charge distribution within the molecule. This involves calculating the vector sum of the individual bond moments within the molecule.

3. What is the formula for calculating the dipole moment?

The formula for calculating the dipole moment is μ = Qr, where μ is the dipole moment, Q is the magnitude of the charge, and r is the distance between the charges.

4. What factors affect the dipole moment of a molecule?

The dipole moment of a molecule is affected by the magnitude and direction of the individual bond moments, as well as the overall geometry and charge distribution of the molecule.

5. How is the dipole moment related to the dielectric constant of a substance?

The dipole moment is directly proportional to the dielectric constant of a substance. The higher the dipole moment, the higher the dielectric constant, indicating a stronger ability of the substance to polarize in an electric field.

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