Calculating divergence as a function of radius

[Caleb Jones]

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Hey guys

Question:
Calculate the divergence as a function of radius for each of the following radially
symmetrical fields in which the magnitude of the field vector:
(a) is constant;
(b) is inversely proportional to the radius;
(c) is inversely proportional to the square of the radius;
(d) is inversely proportional to the cube of the radius.

Im completely stumped on this question...
What I've got so far: (None of this was provided in the question)
Radial field:
V = 1/r² (Vector "r")
Divergence of a spherical Shell:

div F = ∇⋅F

Flux through a spherical shell:
∅ = ∫ E.dA ---> E Constant
∅ = E ∫ dA
∅ = E×4(pi)×r²

Im not sure if I'm on the right path here though

Cheers
Caleb

 

[mfb]

Radial field:
V = 1/r² (Vector "r")

That is a scalar field, not a vector field. A vector field could be ##\vec F = \vec r##, for example.
You'll have to find the correct fields first.

 

[Caleb Jones]

That is a scalar field, not a vector field. A vector field could be ##\vec F = \vec r##, for example.
You'll have to find the correct fields first.

Thank you mfb
How do I find these fields?
Can I just use any symmetric field?
Sorry for my lack of knowledge, this hasn't been explained in lectures or in our lecture notes

 

[mfb]

Can I just use any symmetric field?

You'll need a field that is (a) constant with r, (b) inversely proportional to the radius, and so on. The field I gave as example is proportional to the radius.

 

[rude man]

Follow post #4 to get your 4 fields. His example (field proportional to r) could also be written F = k₁ r with r as the unit vector so F_r = k₁ where F = F_r r.

What is the expression for ∇⋅ F for cylindrical coordinates? Look it up most anywhere. Rest is a gimme.