How Can You Find Vector Potentials for a Magnetic Field?

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SUMMARY

The discussion focuses on finding vector potentials for a magnetic field generated by a long straight wire carrying a current I along the z-axis, represented by the equation \(\mathbf{B} = \frac{\mu_0I}{2\pi} \{\frac{-y}{x^2+y^2} \hat{x} + \frac{y}{x^2 + y^2} \hat{y} \}\). Participants are tasked with deriving two distinct vector potentials that yield this magnetic field and demonstrating that the curl of the difference between these potentials is zero. The primary equation referenced is \(\nabla \times \mathbf{A} = \mathbf{B}\), which is essential for solving the problem.

PREREQUISITES
  • Understanding of vector calculus, specifically curl and cross products.
  • Familiarity with magnetic fields and vector potentials in electromagnetism.
  • Knowledge of Maxwell's equations, particularly the relationship between magnetic fields and vector potentials.
  • Proficiency in mathematical manipulation of partial derivatives.
NEXT STEPS
  • Study the derivation of vector potentials from magnetic fields in electromagnetism.
  • Learn about the mathematical properties of curl and divergence in vector calculus.
  • Explore examples of vector potential calculations for different current configurations.
  • Investigate the implications of gauge invariance in electromagnetic theory.
USEFUL FOR

Students and professionals in physics, particularly those specializing in electromagnetism, as well as educators seeking to enhance their understanding of vector potentials and magnetic fields.

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Homework Statement


A magnetic field of a long straight wire carrying a current I along the z-axis is given by the following expression:

[tex]\mathbf{B} = \frac{\mu_0I}{2\pi} \{\frac{-y}{x^2+y^2} \hat{x} + \frac{y}{x^2 + y^2} \hat{y} \}[/tex]

Find two different potentials that will yield this field. Show explicitly that the curl of the difference between these two potentials vanishes.

Homework Equations


[tex]\nabla \times \mathbf{A} = \mathbf{B}[/tex]

The Attempt at a Solution


I took a cross product to get this system...

[tex] \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} = 0 \\[/tex]

[tex] \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} = \frac{-y}{x^2+y^2} \\[/tex]

[tex] \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} = \frac{y}{x^2+y^2} \\[/tex]

I don't know what to do! Any ideas?
 
Last edited:
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Maybe nobody knows how to do this problem...

But does anybody know how to, in general, find a vector 'A' given its cross product 'B'?
 

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