Average Electric Field and the Electric Dipole Moment

Show that the average electric field over the volume of a sphere is related only to the electric dipole moment of the charge distribution inside the sphere even though the charge distribution may have other multipole moments. Figure 1



QUESTIONS

(Score is number right minus number wrong.)

We start by proving a Lemma: Show that the average electric field inside a sphere due to a SINGLE charge q at an arbitrary point P inside the sphere is the same as the field at that point due to a uniformly charged sphere with -q distributed uniformly throughout the sphere.

On the one hand...

The weights (w's) for calculating the average electric field in the sphere using Equation (4) are the differential volume elements.
True
False
Equation (7) is a correct rendering of Equation (4) for the case of a single charge q at point P inside the sphere.
True
False
The "source point" for purposes of specifying the direction of wavy r in Equation (7) is the position of the differential volume element.
True
False
On the other hand, if we remove the single charge and replace it in the sphere with a uniform charge distribution given by Equation (8)...

The "source point" for purposes of specifying the direction of wavy r in applying Equation (3) for a uniform charge distribution is the position of the differential volume element.
True
False
In Equation (9), we should take the positive sign in front of wavy r.
True
False
The average electric field inside a sphere due to a SINGLE charge q at an arbitrary point P inside the sphere is the same as the field at that same point P due to a uniformly charged sphere with -q distributed uniformly throughout the sphere.
True
False
The utility of the Lemma is that it is simple to get the electric field at point P for the posited uniform charge distribution and, hence, to get the average electric field. The result is given by Equation (10).
True
False
Unfortunately, a single charge at position P does not have a dipole moment. There must be two opposite charges separated by a distance for there to be a dipole moment.
True
False
According to Equation (5), a single charge at position P has a dipole moment relative to the center of the sphere.
True
False
For a single charge at point P, the average electric field in the sphere is given by Equation (11).
True
False
Electric fields are vector quantities. We could have had two single charges and added electric fields at every step to yield Equation (12).
True
False
The average electric field over the volume of a sphere is related only to the electric dipole moment of the charge distribution inside the sphere even though the charge distribution may have other multipole moments.
True
False
CONCLUSION: Within matter where the microscopic fields are complicated functions of the positions and orientations of atoms and molecules, we can nevertheless calculate average electric fields by knowing the dipole moments (only) of the charge distributions.
EQUATIONS

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