Dipole moment of polarized sphere

[SallyBieber]

Homework Statement

Consider a polarised sphere of radius R the polarization is given by
P vector = (ar^2 + b) r hat = ( ar+ b/r) r vector
Where a and b are constant

Homework Equations

Find the dipole moment of the sphere

The Attempt at a Solution

I knew that P (polarized)= delta p / delta volume
So to find dipole moment
I'll take : delta p = \int p . dv
I have a solution manual written in it that
P.r vector = Q
So the delta p = Q/4 pi r^2
How they got this equ.
Should i use the polarized equ. That given
Im so confused

Help

 

[SallyBieber]

Is here anyone can help!,,

 

[TSny]

I don't think you need to do any calculation here.

Note that the polarization vector P has a direction that is radially outward at each point of the sphere and the magnitude of P depends only on r.

So, if you consider a small element of volume dV₁ at some point in the sphere at a distance r from the center, the dipole moment of that element will be dp₁ = P₁dV₁.

Now consider a second volume element dV₂ (same size as dV₁) that is at the same distance r from the center but is on the opposite side of the center from dV₁. The dipole moment of that element will be dp₂ = P₂dV₂.

How do the magnitudes of dp₁ and dp₂ compare? How do their directions compare?

What would you get if you add them: dp₁ + dp₂ = ?