Multipole expansion of Vector Potential (A)

In summary, the conversation discusses the proof that the monopole term in the multipole expansion of the vector potential (\vec A) is equal to zero. The proof involves manipulating integrals and using the unit vector (n), current density vector (J), and Legendre's polynomials. The confusion arises when trying to understand why the last term in the proof is zero, but it is eventually explained that this is due to the condition that J is only defined in a certain region of space.
  • #1
Paradoxx
18
0

Homework Statement


So my teacher, as we made the multipole expansion of Vector Potential (\vec A) decided to proof that the monopole term is zero doing something like this:

∫∇'⋅ (J.r'i)dV' = ∮r'iJ ndS' = 0
The first integral, "opening" the nabla: J⋅(∇r'i) + r'i(∇⋅J) this must be equals 0

J = current density vector
n = unit vector

Homework Equations


I didn't get why the last thing (the nabla opened thing) was zero...
He couldn't explain quite well and got confused, then the next day he added one condition to that be zero:
[*] J is defined only in a certain region of space and that's why it's zero.

I didn't get at all...

The Attempt at a Solution



Expanding using Legendre's polinomes the first term comes to: A_0 =mu0 I / {4pir} ∫dl = 0 and I got why, it's because the closed contour.

ps.: Sorry for my bad English :(
 
Last edited:
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  • #2
Sorry, the latex didn't work. Gonna edit this in one minute.
 

1. What is the multipole expansion of vector potential?

The multipole expansion of vector potential is a mathematical technique used to express the vector potential (A) in terms of a sum of simpler functions. It is commonly used in the study of electromagnetism to approximate the behavior of a vector potential in regions far from its source.

2. How is the multipole expansion of vector potential calculated?

The multipole expansion of vector potential is calculated using the formula: A = ∑n=0∞ (1/r)(n+1) ∫[r'nPn(cosθ')] ρ(r') dV', where A is the vector potential, r is the distance from the source, Pn(cosθ') is the Legendre polynomial of degree n, ρ(r') is the source charge density, and dV' is the volume element.

3. What is the purpose of using the multipole expansion of vector potential?

The purpose of using the multipole expansion of vector potential is to simplify the calculation of the vector potential in regions far from its source. It allows us to approximate the behavior of the vector potential using a finite number of terms, making it easier to analyze and understand.

4. Can the multipole expansion of vector potential be used to calculate the magnetic field?

Yes, the multipole expansion of vector potential can be used to calculate the magnetic field. The magnetic field (B) can be derived from the vector potential using the formula: B = ∇ x A, where ∇ is the gradient operator. Therefore, by calculating the multipole expansion of vector potential, we can also determine the behavior of the magnetic field.

5. Are there any applications of the multipole expansion of vector potential outside of electromagnetism?

Yes, the multipole expansion of vector potential has applications in other areas of physics, such as quantum mechanics and fluid dynamics. In quantum mechanics, it is used to describe the magnetic properties of atoms and molecules. In fluid dynamics, it is used to study the behavior of vortices and other flow patterns in fluids.

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