Decompose the E field into conservative and non-conservative parts

[alan123hk]

I'm not trying to prove anything in this discussion thread, I'm just describing a concept and a simple calculation method. If people question or even thinks this is wrong, I'll try to explain it.

I should also mention about that the so-called build-up of charge on the surface of the wire creates an electric field that counteracts and cancels the induced electric field inside the wire. If there is a curl of electric field inside the wire, it cannot be eliminated. In this case, the curl of electric field creates large eddy currents inside the wire, resulting in energy losses (ohmic losses and radiation, etc.). In other words, it can only be said that this cancellation works in a fixed direction and very localized region of space.

 

[hutchphd]

If people question or even thinks this is wrong, I'll try to explain it.

Because we may consider the method incorrect does not necessarilly mean we misunderstand your method.
If we do misunderstand please explain, but please seriously consider the other possibility.

 

[alan123hk]

The description of example 10.12 (https://web.mit.edu/6.013_book/www/chapter10/10.1.html) may be considered wrong, because the induced electric field is generally a time-varying field, so all other electric fields correspondingly generated in the system are time-varying, it cannot be accurately described by Laplace's equation. This is because Laplace's equation only applies to regions that do not contain charge, current, or time-dependent electromagnetic phenomena.

However, this is probably a matter of opinion. Since the author uses Laplace's equation, it has been implied that the rate of change of the current should be a constant to get an accurate answer. Even if it is not constant, as long as the rate of change is low enough or the frequency is low enough, an approximation can be obtained. If the error in this approximation is considered acceptable, then it is a correct calculation.

 

[hutchphd]

The title of the chapter might be a clue.

10.1​

Magnetoquasistatic Electric Fields in Systems of​

PerfectConductors​

 

[alan123hk]

Similarly, in the following equation, if the charge distribution and current are constant or change very slowly, all the electric fields generated by the charges can be approximately described by the conservative fields $~-\nabla ~ \theta~$, and the term $-\frac {\partial A} {\partial t}$ only represents the induced electric field.
$$ \\ E=-\nabla ~ \theta-\frac {\partial A} {\partial t}=E_c+E_a $$ If $-\frac {\partial A} {\partial t}$ to represent the only non-conservative field $E ~$that actually exists. Then it will become as follows. $$E= E_c+E ~~~~~\Rightarrow~~~~~E_c=0$$