What is the derivation of Maxwell's 4th equation for a static electric field?

[Master J]

In my derivation of Maxwell's 4th equation from the empirical Biot Savart law, for a static electric field, I have that the curl of B is equal to the magnetic permeability times the current density.

Now, the source coordinates (i. the circuit) are given by R_1 ( R is a vector). The field coordinates, that is, a point outside the circuit at which the field is measured, are given by R_2. As stated above, the curl of B is equal to the magnetic permeability times the current density, but this current density is evaluated at R_2.

This is of course zero. So this means that the curl of B is zero everywhere except on the circuit itself (in the wires). Am I correct? The derivation is the one I was thought in my Electromagnetism class, so I don't doubt it is correct.

Could someone perhaps expand on this for me. Am I correct?

 

[yungman]

In my derivation of Maxwell's 4th equation from the empirical Biot Savart law, for a static electric field, I have that the curl of B is equal to the magnetic permeability times the current density.
You meant magnetic field?
Now, the source coordinates (i. the circuit) are given by R_1 ( R is a vector). The field coordinates, that is, a point outside the circuit at which the field is measured, are given by R_2. As stated above, the curl of B is equal to the magnetic permeability times the current density, but this current density is evaluated at R_2.

This is of course zero. So this means that the curl of B is zero everywhere except on the circuit itself (in the wires). Am I correct? The derivation is the one I was thought in my Electromagnetism class, so I don't doubt it is correct.

Could someone perhaps expand on this for me. Am I correct?

the current density is at the source:

$$\nabla \vec B = \mu_0 \vec J \hbox { better yet using integral form } \int_C \vec B \cdot d\vec S = \mu_0 I$$

Means the magnetic field is generated by the current I passing through a wire etc. The magnetic field is circulating around the current carrying wire. so even if you have a field point destinated by $\vec R_2$, all you do is to put into this formula

$$d\vec B_{(R_2)} \;=\; \frac {\mu_0 I}{4\pi}\;\; \frac {d\vec l X \vec R_2 }{R_2^3}$$

For any point ( as you called R_1) and the magnetic field from the complete wire is

$$\vec B_{(R_2)} \;=\; \frac {\mu_0 I}{4\pi}\;\; \int _C \frac {d\vec l X \vec R_2 }{R_2^3}$$

I don't use R_1 because the wire that carry current is always the source, you cannot take the magnetic field at R_1 and find the magnetic field in R_2.