Range of validity of Maxwell's equations with charges

[fluidistic]

Maxwell's equations with charges can be written as the following (in the cgs system):

$$\frac{\partial \vec E}{\partial t} =c \vec \nabla \wedge \vec B-4\pi \vec J$$.
$$\frac{\partial \vec B}{\partial t} =-c \vec \nabla \wedge \vec E$$.
$$\vec \nabla \cdot \vec E =4 \pi \rho$$.
$$\vec \nabla \cdot \vec B =0$$.

If I'm right, these equations are valid only where charges are present. It means, almost nowhere! For instance consider a positively charged table. $$\rho(t,\vec x)$$ is of course not continuous since we're dealing with charges. In other words it is zero everywhere when there's no charge. If we assume the electron has a volume, then the equations would be valid within the volume of the electron. So in an neutral atom, these equations are valid less than 0.001% of the space (only where the proton and the electron are).
I'm just curious if I'm right. In an affirmative case, why do we bother with them and why don't we only use Maxwell's equation in vacuum?
In the latter case, I could imagine having a lot of boundary equations (precisely the places enclosing the charges) and I could only use Maxwell's equation in vacuum, taking into account all the boundary equations not to lose any information about the charges.

 

[Matterwave]

Maxwell's equations are valid even when no charges are present. In that case, you can set $$\rho=0$$ and you can see that the equations are still there, and the E and B fields also interact.

We use this formulation because it's more general than the formulation without charges.

Because the equations are differential equations, even if some terms equal zero, the E and B fields need not equal zero. Just like if dx/dt=0 it doesn't mean x=0 for all time.

 

[fluidistic]

Maxwell's equations are valid even when no charges are present. In that case, you can set $$\rho=0$$ and you can see that the equations are still there, and the E and B fields also interact.

We use this formulation because it's more general than the formulation without charges.

Because the equations are differential equations, even if some terms equal zero, the E and B fields need not equal zero. Just like if dx/dt=0 it doesn't mean x=0 for all time.

Ah ok, thanks.
I'm getting it. I feel I'm learning a lot, it's incredibly nice!

 

[Pengwuino]

You need to consider that charges create fields throughout your space, not simply where the charges are. Infact, in classical EM, the fields diverge at the actual charges. For example your coulomb field blows up as your field point approaches your charges.

When you formulate a problem, you define a space you're working in. Maxwell's equations will be valid in the entire space. You don't have a new equation for every infinitesimal volume in the space, it's simply one set of equations for the whole space given some charge distribution inside.

Think of a gravitational potential. It's defined for your entire volume of space, not just where the mass is.

 

[sardar]

I can confirm that your understanding of the range of validity of Maxwell's equations with charges is correct. These equations are only valid in regions where charges are present, which is a very small portion of the overall space. This is because the equations describe the behavior of electric and magnetic fields, which are created by charges.

However, even though charges may only exist in a small portion of space, their effects can still be felt in the surrounding regions. This is why Maxwell's equations with charges are still important and useful in understanding the behavior of electromagnetic fields. They allow us to predict and explain the behavior of these fields in the presence of charges.

Additionally, we cannot simply use Maxwell's equations in vacuum to describe the behavior of electromagnetic fields in the presence of charges. This is because the presence of charges changes the behavior of the fields and therefore, the equations must be modified to take this into account. Using only Maxwell's equations in vacuum would not accurately describe the behavior of electromagnetic fields in the presence of charges.

In summary, while Maxwell's equations with charges may only be valid in a small portion of space, they are still crucial in understanding the behavior of electromagnetic fields and cannot be replaced by using only Maxwell's equations in vacuum.