Vector Potential: How to Find it?

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SUMMARY

The discussion focuses on finding the vector potential for the vector field V = x²i + 3xz²j - 2xzk, which has a divergence of zero. Participants highlight that the vector potential can be expressed as the curl of another vector, emphasizing the concept of gauge choice. Unlike scalar potential, which has a single degree of freedom, vector potential allows for multiple degrees of freedom, permitting the addition of a position-dependent function with a zero gradient. This results in additional constants when determining the vector potential.

PREREQUISITES
  • Understanding of vector calculus, specifically divergence and curl operations.
  • Familiarity with the concepts of scalar and vector potentials in physics.
  • Knowledge of gauge invariance and its implications in vector fields.
  • Basic integration techniques involving multiple variables.
NEXT STEPS
  • Study the properties of vector fields and their potentials in electromagnetism.
  • Learn about gauge transformations and their applications in physics.
  • Explore advanced integration techniques for functions of multiple variables.
  • Investigate the mathematical foundations of curl and divergence in vector calculus.
USEFUL FOR

Students and professionals in physics, particularly those studying electromagnetism, as well as mathematicians and engineers working with vector fields and potentials.

PrinceOfDarkness
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Vector V=x^2i+3xz^2j-2xzk
The divergence of this vector is zero. So it can be expressed as the curl of a vector. I have to find that vector, which is also called the vector potential.
But I don't know how to find it. When I have to find the scalar potential, then it is easier to equate components and then integrate. But in this case, if I equate components, I get two variables of integration. How can I integrate then? Is it possible to find a vector potential like this?
 
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PrinceOfDarkness said:
But in this case, if I equate components, I get two variables of integration. How can I integrate then? Is it possible to find a vector potential like this?

It sounds like you're doing the right thing. What you're getting is what the physicists call a "choice of gauge" for the potential.

For the scalar potential, there was only one degree of freedom, and you know that that amounted to just an aribitrary offset. That is, if you add a constant to the potential, the potential still gives the same force.

For the vector potential, there are more than one degree of freedom, and this means that instead of just adding a constant, you can actually add a function of position. The function must be one that has a gradient of zero, but there are plenty of those.

And so, when you look for the vector potential, you're going to get more constants.

Carl
 

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