Understanding Bound Charges and Their Mathematical Derivation

[aaaa202]

Upon reading about bound charges I stumbled on something I didn't quite understand. It is not a physical thing but purely a mathematical thing.

In the attached section my book wants to take the gradient:

∇'(1/r)

with respect to the source coordinates, r'. Now, can someone by inspection of the attached file tell me what these source coordinates represent. Are they they coordinates of a point inside some charge distribution with respect to a fixed point inside the distribution? Would that then mean that in vector notation:

r = R + r'

where R is the distance from P to the reference point inside the distribution?

And from all that can someone tell me how you would differentiate ∇'(1/r) with respect to
r' to get the answer in the bottom of the attached file? :)

thanks

 

[tiny-tim]

hi aaaa202!

… can someone by inspection of the attached file tell me what these source coordinates represent. Are they they coordinates of a point inside some charge distribution with respect to a fixed point inside the distribution?

no, (the diagram should say so, but doesn't …) they're the coordinates of the point marked "P" (which isn't the name of the point, it's the dipole moment density vector ) wrt a fixed origin (whose position doesn't matter)

… how you would differentiate ∇'(1/r) with respect to r' to get the answer in the bottom of the attached file? :)

should be easy now

 

[aaaa202]

I'm still a little confused on how r depends on r'. If R is the distance to the origin used for the coordinates r' isn't then, as I said:

r = R + r'

? :)

 

[tiny-tim]

let's see …

in that integral, r is the outside point, and is fixed (a constant)

r is explained as the distance from r to r',

so r² (the denominator) = (r - r')²

(the notation they're using is very misleading )