Divergence and curl of vector field defined by \vec A = f(r)vec r

In summary, a vector field defined by <i>vec A = f(r)vec r</i> is a mathematical function that assigns a vector to each point in space based on the magnitude and direction of the position vector <i>vec r</i> multiplied by a scalar function <i>f(r)</i>. Divergence and curl are important measures of how a vector field changes from point to point, with divergence representing net flow and curl representing rotation or circulation. The divergence of a vector field <i>vec A = f(r)vec r</i> can be calculated using the formula: <i>div(vec A) = (1/r^2)(d/dr)(r^2A)</i>, and the
  • #1
FourierX
73
0

Homework Statement


A vector field is defined by A=f(r)r
a) show that f(r) = constant/r^3 if [tex]\nabla[/tex]. A = 0
b) show that [tex]\nabla[/tex]. A is always equal to zero

Homework Equations


divergence and curl relations

The Attempt at a Solution


I tried using spherical co-ordinates to solve this. But I am not sure if i am totally right.
 
Last edited:
Physics news on Phys.org
  • #2
I would also use spherical coordinates on this problem.

I cannot say if you are right since I haven't seen any of your work. I'll need to see some work if you want more specific advice.
 
  • #3
FourierX said:

Homework Statement


A vector field is defined by A=f(r)r
a) show that f(r) = constant/r^3 if [tex]\nabla[/tex]. A = 0
b) show that [tex]\nabla[/tex]. A is always equal to zero

Homework Equations


divergence and curl relations

The Attempt at a Solution


I tried using spherical co-ordinates to solve this. But I am not sure if i am totally right.

Certainly parts (a) and (b) can not be consistent as you have written them... i suppose part (b) should be a curl not a divergence?
 
  • #4
here is what I've done and where i got stuck:

[tex]\nabla[/tex]A = [tex]\frac{1}{r^{2}}[/tex] [tex]\frac{\partial}{\partial r} (r^{2}f(r))[/tex]

on simplifying this i got:

f(r) = [tex]\frac{-1}{2} r f'(r)[/tex]

then i integrated with a hope to get an expression for f(r) but did not end up with what i needed i.e. f(r) = constant/ r^{3}
 
  • #5
Well the way you interpreted it, you have [tex]\mathbf{A}=f(r)\hat{r}[/tex], but the way the question is stated it is [tex]\mathbf{A}=f(r)\mathbf{r}=[/tex], or [tex]\mathbf{A}=f(r)r\hat{r}[/tex]. This extra factor of r should give the answer required.
 

1. What is a vector field defined by vec A = f(r)vec r?

A vector field defined by vec A = f(r)vec r is a mathematical function that assigns a vector to each point in space. The vector at each point is determined by the magnitude and direction of the vector vec r (the position vector of that point) multiplied by a scalar function f(r) that depends on the distance from the origin.

2. What is the significance of divergence and curl in vector fields?

Divergence and curl are two important measures of how a vector field changes from point to point. Divergence measures the net flow of a vector field out of a given region, while curl measures the rotation or circulation of the vector field around a given point. These concepts are essential for understanding the behavior of fluid flow, electromagnetic fields, and other physical phenomena.

3. How do you calculate the divergence of a vector field vec A = f(r)vec r?

The divergence of a vector field vec A = f(r)vec r can be calculated using the formula: div(vec A) = (1/r^2)(d/dr)(r^2A), where r is the distance from the origin and A is the scalar function that determines the magnitude of the vector at each point. This formula can be simplified for specific forms of f(r), such as f(r) = 1/r for a point source.

4. How is the curl of a vector field vec A = f(r)vec r related to the gradient?

The curl of a vector field vec A = f(r)vec r is related to the gradient of the scalar function f(r) by the formula: curl(vec A) = (1/r)(d/dr)(rA) - (A/r^2)vec r. This shows that the curl can be expressed in terms of the gradient of f(r) and the position vector vec r. This relationship is known as the curl theorem.

5. What are some applications of divergence and curl in real-world problems?

Divergence and curl are used extensively in physics and engineering to analyze and solve problems related to fluid flow, electromagnetism, and other fields. For example, in fluid dynamics, the divergence and curl of the velocity field can be used to calculate the rate of mass flow and the vorticity of the fluid, respectively. In electromagnetism, the divergence and curl of the electric and magnetic fields can be used to determine the presence of charges and currents, and to predict the behavior of electromagnetic waves. These concepts also have applications in other areas such as weather forecasting, image processing, and optimization techniques.

Similar threads

  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
13
Views
2K
  • Advanced Physics Homework Help
Replies
26
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
914
  • Advanced Physics Homework Help
Replies
4
Views
699
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
10
Views
1K
  • Advanced Physics Homework Help
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
12
Views
2K
Back
Top