Vector potential with current density

In summary: So the potential at z is just the sum of all the potentials at all the other observation points weighted by their distances to z.I don't think so. You can get the answer from symmetry considerations. Note that the current ##\vec{j}(r') d^3r'## in a small volume element makes an infinitesitmal contribution to the vector potential at ##r## of amount ##d\vec{A}(r)##, and ##d\vec{A}(r)## has the same direction as ##\vec{j}(r')##. Do you see that each nonzero value of ##\vec{j}(r
  • #1
Faust90
20
0

Homework Statement


Hey,
I got the current density [itex]\vec{j}=\frac{Q}{4\pi R^2}\delta(r-R)\vec{\omega}\times\vec{r}[/itex] and now I should calculate the vector potential:
\vec{A}(\vec{r})=\frac{1}{4\pi}\int\frac{j(\vec{r})}{|r-r'|}.


Homework Equations


The Attempt at a Solution


here my attempt till now:
http://phymat.de/physics.png [Broken]
I'm really not sure how to go on now. Is this right what I wrote there?
 
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  • #2
In the second equation of the notes, should the vector r in the numerator have a prime?

[EDIT: Also, do [itex]\theta[/itex] and [itex]\phi[/itex] refer to the unprimed position or the primed position?]
 
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  • #3
TSny said:
In the second equation of the notes, should the vector r in the numerator have a prime?

[EDIT: Also, do [itex]\theta[/itex] and [itex]\phi[/itex] refer to the unprimed position or the primed position?]

Hi,

thanks for your answer. I think they should all have primes.

Edit: I should only calculate the potential A(r) on the z-axis (for r=z e_z). But I don't know how this can be helpful.
 
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  • #4
Faust90 said:
I should only calculate the potential A(r) on the z-axis (for r=z e_z).

Ahh. If ##\vec{\omega}## is also along the z-axis, then life is looking Wunderbar!
 
  • #5
TSny said:
Ahh. If ##\vec{\omega}## is also along the z-axis, then life is looking Wunderbar!

Hi, thanks,
Yes Omega is also along the z-Axis, but I don't know how this could help me. Didn't I have to solve the integral first?
 
  • #6
Faust90 said:
Hi, thanks,
Yes Omega is also along the z-Axis, but I don't know how this could help me. Didn't I have to solve the integral first?

I don't think so. You can get the answer from symmetry considerations. Note that the current ##\vec{j}(r') d^3r'## in a small volume element makes an infinitesitmal contribution to the vector potential at ##r## of amount ##d\vec{A}(r)##, and ##d\vec{A}(r)## has the same direction as ##\vec{j}(r')##.

Do you see that each nonzero value of ##\vec{j}(r') d^3r'## is parallel to the xy plane? So, the total value of ##\vec{A}(r)## must be parallel to the xy plane at any observation point ##r##.

For this problem the observation point is at some z on the z-axis. Note that the current distribution ##\vec{j}(r')## is axially symmetric about the z axis. So ##\vec{A}(z)## would have to point perpendicular to the z-axis and yet be rotationally invariant about the z-axis. There is only one possibility for the value of ##\vec{A}## at z.
 

What is the definition of vector potential with current density?

The vector potential with current density, also known as the magnetic vector potential, is a mathematical concept used in electromagnetism to describe the magnetic field generated by a current-carrying wire or a moving charged particle. It is a vector quantity that represents the strength and direction of the magnetic field at a given point in space.

How is the vector potential with current density related to the magnetic field?

The vector potential with current density is related to the magnetic field through the equation B = ∇ × A, where B is the magnetic field, A is the vector potential, and ∇ × is the curl operator. This means that the magnetic field at a point is equal to the curl of the vector potential at that point.

What is the importance of the vector potential with current density in electromagnetism?

The vector potential with current density plays a crucial role in describing the behavior of magnetic fields in electromagnetism. It allows us to calculate the magnetic field at any point in space, and it is also used in the calculation of electromagnetic forces and energy.

How is the vector potential with current density calculated?

The vector potential with current density can be calculated using the Biot-Savart law, which relates the magnetic field at a point to the current density. It can also be calculated using Maxwell's equations, specifically the equation ∇ × A = μ0J, where μ0 is the permeability of free space and J is the current density.

What are some real-world applications of the vector potential with current density?

The vector potential with current density has many practical applications in various fields, such as electrical engineering, physics, and geophysics. It is used in the design of electric motors, generators, and transformers, as well as in the study of Earth's magnetic field and its variations. It is also essential in the development of technologies such as MRI machines and particle accelerators.

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