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I've got three physics equations in mind which seem (to me) to contradict eachother, using a simple case of charge(s) in a static electric field. If someone can give an explanation as to what I'm missing that would be much appreciated. The equations:

Maxwell-Amperes circuital law:

$\nabla \times B = \mu_0 (J + \epsilon_0 \frac{\partial{E}}{\partial{t}}) $

Ohms law:

$J = \sigma E$

And finally, Lorentz Force Law (in absence of magnetic field) (along with F = ma):

$F = qE$

Starting with the equation for Coloumb force, this tells me charges should accelerate in the presence of an electric field. Assuming current is proportional to velocity of charges, this suggests that the time derivative of current would be proportional to electric field strength.

Now for ohms law, it clearly states that current (density) is proportional to Electric field strength. I'm guessing this refers to a steady state due to material properties?

Finally, for Maxwell-Amperes law, assuming no magnetic field it suggests current density is proportional the time derivative of electric field, so we have another apparent discrepancy - assuming my above reasoning applies.

The apparent discrepancy between Maxwell and Coloumb laws are what I'm most interested in. So, what am I missing in my logical reasoning? Any insights will be very much appreciated.

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  • $\begingroup$ Do not insert your equations as pictures. Especially when your equations are this simple. Please just use MathJaX instead. $\endgroup$ Jun 26, 2023 at 15:21
  • $\begingroup$ @naturallyInconsistent OK, can you explain why? $\endgroup$
    – Christian
    Jun 26, 2023 at 15:23
  • $\begingroup$ The image links very easily cause link rot. And then nobody can read what your question was asking. MathJaX is much more able for search engines to find. It is a win on all the important fronts. $\endgroup$ Jun 26, 2023 at 15:24
  • $\begingroup$ @naturallyInconsistent OK I will edit it. $\endgroup$
    – Christian
    Jun 26, 2023 at 15:25

1 Answer 1

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Coloumb

The name is Coulomb.

$\vec F=q\vec E$

This is NOT called the Coulomb force. You should call it the Lorentz force law in the absence of magnetic field.

And this is actually the wrong force law to use here. More later.

Now for Ohms law, it clearly states that current (density) is proportional to Electric field strength. I'm guessing this refers to a steady state due to material properties?

Yes.

Finally, for Ampère-Maxwell's law, assuming no magnetic field it suggests current density is proportional the time derivative of electric field, so we have another apparent discrepancy - assuming my above reasoning applies.

Your reasoning does not apply.

this tells me charges should accelerate in the presence of an electric field. Assuming current is proportional to velocity of charges, this suggests that the time derivative of current would be proportional to electric field strength.

You correctly noted that Ohm's Law is really a steady state condition. This means that, in Newton's 2nd Law, it would correspond to the terminal velocity case: $$\tag1\text{N2L}:\qquad\vec F=q\vec E-\vec f_\text{collisions}=\vec0$$ And in this way, because you were using the wrong force law, your correct reasoning led you to a wrong rabbit hole.

Finally, what you really know, is that once the steady state conditions are reached, $\partial_t\vec E=\vec0$, so that in Ampère-Maxwell's law, you cannot have the current density be proportional to the time derivative of the electric field. Instead, what it says is that the steady state circuit loop must produce a magnetic field. Which is necessary for correct operation and understanding of the circuit.

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  • $\begingroup$ Thankyou, I've modified my question to reflect your suggestions. I'm thinking about charges in a vacuum, rather than in a circuit as I feel it's a simpler case. Am I right that the time differential of 'current' should be proportional to electric field strength (as it results in an acceleration of charges)? If so, why does this not seem to be reflected in Maxwells equations? $\endgroup$
    – Christian
    Jun 26, 2023 at 15:51
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    $\begingroup$ You cannot use Ohm's Law in a vacuum. $\endgroup$ Jun 26, 2023 at 15:52
  • $\begingroup$ Happy with your answer regarding ohms law, more interested in the other two. $\endgroup$
    – Christian
    Jun 26, 2023 at 15:53
  • $\begingroup$ I am not sure what you are trying to get. The correct place for it to appear, and it does appear, is in the Lorentz force law. And the other equation is Ampère-Maxwell's law, and what is there to say, to connect, between those concepts. I fear you are trying to tangle yourself, and I am not keen on going down rabbit holes. $\endgroup$ Jun 26, 2023 at 16:00
  • $\begingroup$ Ultimately I am trying to understand how to get from Maxwells equations to deriving currents, however I'm trying to iron out some inconsistencies in my understanding first. $\endgroup$
    – Christian
    Jun 26, 2023 at 16:08

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