The Relation between the integral and differential form of Amperes Law

[Zook104]

The integral form of Ampere's law in vacuum is

∫B$\cdot$dl=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

 

[mfb]

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.

 

[Zook104]

I am sorry but I don't understand what you mean?

 

[mfb]

Can you be more specific where the problem is?
Alternatively, can you show your previous attempts?

 

[paranormal]

(a) The differential form of Ampere's law can be obtained by using the relation between I and J, which states that the current density J is equal to the current I divided by the cross-sectional area A through which the current is flowing. This can be written as J=I/A. Substituting this into the integral form of Ampere's law, we get:

∫B\cdotdl=μ_{0}(I/A)A

Simplifying, we get:

∫B\cdotdl=μ_{0}I

This is the differential form of Ampere's law, which states that the magnetic field B is equal to the permeability of free space μ_{0} multiplied by the current density J.

(b) The displacement current density J_{d} is a term that accounts for the changing electric field in a region, and is defined as J_{d} = ε_{0} dD/dt, where ε_{0} is the permittivity of free space and dD/dt represents the rate of change of the displacement field D.

In the presence of a time-varying electric field, the displacement current density modifies the differential form of Ampere's law by adding an additional term:

∫B\cdotdl=μ_{0}(J+J_{d})

This modified form takes into account the contribution of the changing electric field to the magnetic field, and is necessary for accurately describing electromagnetic phenomena. It is known as the modified Ampere's law.