Exploring Griffiths' Assumptions on Current & Charge Distribution

[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...

 

[paranormal]

I would like to begin by acknowledging the importance of Griffiths' assumptions on current and charge distribution in the field of electrodynamics. These assumptions play a crucial role in our understanding of electromagnetic phenomena and are widely accepted in the scientific community.

Regarding your question, it is important to note that Griffiths is not assuming that the charge and current densities are zero everywhere in space, but rather that they are constant within a specific domain. This domain can be defined by the coordinate system x', y', z', where the charge and current distributions are different from zero. In this case, the divergence and rotor of the charge and current densities will be zero within this domain, as they do not vary with respect to the coordinates x, y, z.

Moreover, Griffiths does not use two coordinate systems to describe the charge and current distributions. He uses one coordinate system, but it is a different one from the one used to describe the electric and magnetic fields. This distinction is important because it allows us to simplify the mathematical calculations and analyze the problem more easily.

As for the references you have mentioned, it is worth noting that Jackson's statement about divJ being zero in steady-state magnetic current is consistent with Griffiths' assumptions. In this case, the current density is constant in time and therefore does not depend on the coordinates x, y, z.

In conclusion, Griffiths' assumptions on current and charge distribution are well-supported and have been confirmed by numerous experiments and observations. While it is important to question and critically analyze scientific theories and assumptions, in this case, it seems that the assumptions made by Griffiths are valid and have been widely accepted in the scientific community.