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Given the electric field defined by: $$\vec {E(\vec r)}= x \hat {x} $$ There is no obvious contradiction with the classic electro-magnetic theory. But: $$\vec \nabla \cdot \vec E = 4 \pi \rho(\vec r) = 1 \Rightarrow \rho=\dfrac {1}{4\pi}$$ Now, this charge density is constant throughout the space, and obviously isotropic. Yet, the field created has certain direction, and it varies in magntitude throughout the space, both last properties are in contradiction to the fact that the charge density is isotropic. Also, other fields, such as: $$\vec E_y = y \hat y $$ will lead to the same charge density function.

I do have some intuition to what is the problem: I think the irrelevancy of such fields and charge densities is the root of the problem. Although they satisfy some fundemental equations of classical physics - they cannot exist in our universe.

Is there another explanation to this?

SchrodingersCat
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Ranc
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  • The electric field of the uniformly charged space is not only isotropic, but 0 everywhere. – Rol Sep 26 '15 at 18:07
  • It is very intuitive, but i can't figure it out out of the equations. – Ranc Sep 26 '15 at 19:44
  • @Rol: If that's true, then the electric field doesn't satisfy Gauss's Law, since we'll have $\vec{\nabla} \cdot \vec{E} = 0 \neq \rho/\epsilon_0$. – Michael Seifert Nov 08 '21 at 18:35
  • @MichaelSeifert I guess I fell for this: https://physics.stackexchange.com/questions/39654/paradox-with-gauss-law-when-space-is-uniformly-charged-everywhere – Rol Nov 10 '21 at 04:55

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It is true that given an electric field, then you can define uniquely the charge density that created it, by Gauss' law, as you have done. But the converse is not true: given a charge density you cannot define uniquely the electric field that it will create since you have to solve a differential equation (again Gauss' law) to do that and each differential equation gives a unique solution only if it is accompanied by some boundary conditions. So, the same charge density can produce different electric fields for different boundary conditions.

user3257624
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