Are Griffiths' Assumptions on Charge and Current Distribution Justified?

[nikolafmf]

I will refer to Griffiths' textbook Introduction to Electrodynamics, Third Edition.

On page 70 he calculates divergence of E and implicitely assumes that divergence of rho is 0, where rho is charge density distribution. On page 223 he calculates rotB and says that rotJ = 0, where J is current density distribution. He says that rho and J depend only on x', y', z', but not on x, y, z. That is the reson why their divergences or rotors are 0.

My question is, aren't this x', y' and z' just equal x, y and z on the specific domain where charge, or current, distribution densities are diferent from zero? Griffiths uses only one coordinate system, not two! In that case, should't divergence and rotor of rho or J be diferent from zero at that domain, because actualy they depend on x, y, z?

I have one more reason to suspect that Griffiths is right. In Jackson's Classical Electrodynamics, Third Edition, on page 179 it is said that divJ is zero as a result that we analyze steady state magnetic current, not because J depends on x', y', z' and not on x, y, z. In clasical Electrodynamic of Greiner, first English edition, on page 193 it is said that divJ = 0 as a result that problem is magnetostatics one.

Any comment on this will be appresiated.
Nikola

 

[dingo_d]

primed and unprimed variables are a part of the same coordinate system. Unprimed just means that it's the coordinate where you look at the specific field or current density that emanates from primed coordinates.

And $$\nabla\times \jmath=0$$ is because, basically, he is differentiating a constant (it's like if you try to find $$\frac{d y}{dx}$$ - it's zero because y doesn't depend on x). I mean, the current is not a constant, but he's differentiating with respect to something that current density doesn't depend on...

 

[nikolafmf]

primed and unprimed variables are a part of the same coordinate system. Unprimed just means that it's the coordinate where you look at the specific field or current density that emanates from primed coordinates.

And $$\nabla\times \jmath=0$$ is because, basically, he is differentiating a constant (it's like if you try to find $$\frac{d y}{dx}$$ - it's zero because y doesn't depend on x). I mean, the current is not a constant, but he's differentiating with respect to something that current density doesn't depend on...

But here, I think, exists a map between primed and unprimed coordinates as this:

x'->x, y'->y, z'->z in the region where charge or current density is different from zero. Then, any function J (x', y', z') = J (x, y, z) in that region, doesn't it? If so, then current density does depend on x, y, z.

 

[dingo_d]

Ummm I don't think so.

You have at the beginning the Biot-Savart law. You want to find the magnetic field at point P (x,y,z) that comes from a current density $$\jmath$$ in point A(x', y', z') in some small volume dV.

The rest follows from the explicit formula...

True J(x',y',z')=J(x,y,z) but only if you are looking at the same point (A=P). And here you want to find the field at the point P that is some distance r away...

Because if you'd look at the same spot the distance would be zero and the whole integral would be zero...