Vector magnetic potential of current sheet

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SUMMARY

The discussion centers on the calculation of the vector magnetic potential, \(\mathbf A\), for an infinite current sheet using the formula \(\mathbf A=\frac{\mu_0}{4\pi}\int_{V'}\frac{\mathbf J}{R}dv'\). Participants confirm that the integral does not converge due to the infinite extent of the current sheet. However, they establish that calculating the curl of \(\mathbf A\) using \(\nabla\times\mathbf A\) yields valid results. It is concluded that while the potential \(\mathbf A\) cannot be defined globally for an infinite current sheet, a local potential can be defined.

PREREQUISITES
  • Understanding of vector calculus, particularly curl operations.
  • Familiarity with magnetic vector potential and its physical significance.
  • Knowledge of Maxwell's equations and their implications for electromagnetic fields.
  • Basic concepts of charge distribution and electric potential in electrostatics.
NEXT STEPS
  • Study the derivation and implications of the magnetic vector potential in electromagnetic theory.
  • Explore the concept of local potentials in electrostatics and magnetostatics.
  • Learn about the mathematical treatment of infinite charge distributions and their effects on potential calculations.
  • Investigate the application of the Biot-Savart law in calculating magnetic fields from current distributions.
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism, particularly those interested in advanced topics related to magnetic fields and potentials in infinite systems.

daudaudaudau
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Hi. Say I have an infinite sheet of current. My book gives the following formula for the vector magnetic potential

<br /> \mathbf A=\frac{\mu_0}{4\pi}\int_{V&#039;}\frac{\mathbf J}{R}dv&#039;<br />

But when I do the integral, it doesn't converge. However, if I calculate \nabla\times\mathbf A, i.e. move the \nabla\times inside the integral, it works out fine. Is it really impossible to calculate \mathbf A for an infinite current sheet? I have the same problem if I try to calculate the potential V of an infinite sheet of charge, but for the electric field \mathbf E it works out fine.
 
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No. The potential of an infinite sheet does not converge at infinity. But you can define a local potential.
 

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