Multipole expansion of Vector Potential (A)

Click For Summary
SUMMARY

The discussion focuses on the multipole expansion of the vector potential (\vec A) in electromagnetism, specifically addressing why the monopole term is zero. The proof involves the integral ∫∇'⋅ (J.r'i)dV' and its transformation into a surface integral, which equals zero due to the properties of the current density vector (J) and the defined region of space. The participant clarifies that the zero result arises from the closed contour integral, as demonstrated using Legendre polynomials.

PREREQUISITES
  • Understanding of vector calculus, specifically divergence and surface integrals.
  • Familiarity with electromagnetism concepts, particularly vector potential and current density.
  • Knowledge of multipole expansion techniques in physics.
  • Basic proficiency in using Legendre polynomials in mathematical physics.
NEXT STEPS
  • Study the derivation and applications of multipole expansions in electromagnetism.
  • Learn about the properties of vector fields and their integrals in vector calculus.
  • Explore the role of Legendre polynomials in solving physical problems.
  • Investigate the implications of current density distributions in different regions of space.
USEFUL FOR

Students of physics, particularly those studying electromagnetism, as well as educators seeking to clarify concepts related to vector potentials and multipole expansions.

Paradoxx
Messages
18
Reaction score
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:
Physics news on Phys.org
Sorry, the latex didn't work. Gonna edit this in one minute.
 

Similar threads

Undergrad Origin of coordinates in multipole expansion
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Vector Potential Multipole Expansion
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Confused about multipole expansion of vector potential
  • · Replies 4 ·
Replies
4
Views
1K
Multipole expansion - small problem
  • · Replies 3 ·
Replies
3
Views
2K
Deriving magnetic dipole moment from multipole expansion
  • · Replies 1 ·
Replies
1
Views
3K
Using Legendre Polynomials in Electro
  • · Replies 3 ·
Replies
3
Views
3K
Multipole Expansion: Quadrupole Moment Calculation
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Expanding around 0 for multipole expansion
  • · Replies 4 ·
Replies
4
Views
924
Force acting between bodies - using multipole expansion
  • · Replies 7 ·
Replies
7
Views
3K
Electrostatic polarization of an axially symmetric conductor
  • · Replies 8 ·
Replies
8
Views
3K