The Relation between the integral and differential form of Amperes Law

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SUMMARY

The integral form of Ampere's Law in vacuum is expressed as ∫B·dl = μ₀I. To derive the differential form, one must relate the current I to the current density J, ignoring displacement current. The displacement current density Jₑ is defined in terms of the displacement field D, which modifies the differential form of Ampere's Law by incorporating the effects of changing electric fields. This discussion emphasizes the importance of understanding the relationship between integral and differential forms of Ampere's Law in electromagnetic theory.

PREREQUISITES
  • Understanding of Ampere's Law and its integral form
  • Familiarity with differential calculus and vector calculus
  • Knowledge of electromagnetic fields, specifically magnetic field B and electric displacement field D
  • Concept of current density J and displacement current density Jₑ
NEXT STEPS
  • Study the derivation of the differential form of Ampere's Law from its integral form
  • Learn about the concept of displacement current density Jₑ and its implications in electromagnetic theory
  • Explore vector calculus techniques, particularly the curl operator, in the context of electromagnetic fields
  • Investigate Maxwell's equations and their applications in various electromagnetic scenarios
USEFUL FOR

Students and professionals in physics, particularly those focusing on electromagnetism, electrical engineers, and anyone studying Maxwell's equations and their applications in theoretical and applied physics.

Zook104
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The integral form of Ampere's law in vacuum is

∫B\cdotdl=μ_{0}I

(a) Using the relation between I and J, obtain the differential form of Ampere's
law. You may ignore any displacement current.

(b)Define the displacement current density J_{d} in terms of the displacement
field D and show how it modifies the differential form of Ampere's law.

My attempts at this have circular and achieved no useful answers. So all and any help would be greatly appreciated :D
 
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You can consider special paths for the integration - like squares or circles - and then let their size go to zero. The interesting part is how you get rot(B) out of that limit.
 
I am sorry but I don't understand what you mean?
 
Can you be more specific where the problem is?
Alternatively, can you show your previous attempts?
 

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