How Does Magnetostatics Explain Zero Divergence and Vector Potentials?

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SUMMARY

This discussion focuses on the principles of magnetostatics, specifically addressing the zero divergence of the vector potential \( A \) in static fields and the relationship between the vector potential \( A \) and magnetic flux density \( B \). The user seeks clarification on the equations \( \nabla \cdot A = 0 \) and \( \nabla^2 A = -\mu_0 j \), which lead to the conclusion that \( \nabla \times B = \mu_0 j \). The Biot-Savart Law is also referenced for calculating magnetic flux density due to current-carrying wires.

PREREQUISITES
  • Understanding of vector calculus, particularly divergence and curl operations.
  • Familiarity with magnetostatics concepts, including vector potentials and magnetic fields.
  • Knowledge of the Biot-Savart Law for calculating magnetic fields from current distributions.
  • Basic proficiency in LaTeX for mathematical expressions.
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  • Study the derivation of the Biot-Savart Law in detail.
  • Learn about the implications of the vector potential in electromagnetic theory.
  • Explore the mathematical properties of divergence and curl in vector fields.
  • Investigate the physical significance of magnetic flux density \( B \) in various configurations of current-carrying wires.
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Students and professionals in electromagnetism, physicists studying magnetostatics, and engineers working with electromagnetic fields and applications.

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Hi :)

this is my first post to this forum. I am doing some study in EM
and I've come across some helpful hints on here, to help me through some problems. However i have come across a couple stumbling blocks.
if anyone could give me a couple clues to go about working these out and give me a direction to go in, it would be much appreciated. cheers in Advance.

1) how that for static fields (no time dependece and only currents of finite extent) the divergence potential is zero, i.e., [tex]\nabla\cdot[/tex]A = 0
(note the A is a vector)

2) We found that for static fields the vector potential obeys
[tex]\nabla^2[/tex] A = - [tex]\mu[/tex] oj . Show that from this equation follows that [tex]\nabla\times[/tex] B = [tex]\mu[/tex] oj

again A, B, and j are vectors.

hope my latex works..

thanks again
 
Last edited:
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What is the relationship between A and B?
 
Hi HallsofIvy

Thanks for your reply.

these two are from a four set of question i had been looking at.
The first asked "to compute the magnetic flux density B for a point P due to the electric current I in a long, straight wire (distance between wire and point p is [tex]\rho[/tex]
Using biot-savart law i came up with the following

[tex]\frac{\mu_o I}{2 \pi\rho}[/tex] a

this may not look exactly as what i got as I am still unsure about using Latex.

The second part, asked, to compute the vector potential and magnetic flux density for a point P located between two parallel (straight) wires (distance between the wires is D) carrying electric currents of the same magnitude I in opposite directions.
I completed this and proceeded onto looking at the two i had asked on here, but wasnt sure what to do with those 2. I've been back over the second part however and I am re-working it, think I've made mistake in it's working out.

i assume you were asking the relationship from therre first two parts that i determined?
 

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