B Decompose the E field into conservative and non-conservative parts

[vanhees71]

It's in any textbook on electromagnetism, and you don't need to rely on peer-review, if you calculate it yourself.

The most simple case is a very long coaxial cable. with a battery on one end and a resistor (or a short circuit) on the other. Then (in non-relativistic approximation) you can make the ansatz that $\vec{j}=j \vec{e}_3$ (where the cable is along $\vec{e}_3$) with $j=\text{const}$ along the wires and $0$ outside.

Then you can solve the static Maxwell equations together with the appropriate boundary conditions and get the surface charges along the surfaces as well as the electric and magnetic fields everywhere.

 

[alan123hk]

simple question. In the following system, due to the induced electric field generated by the inductance, there must be only one electric field, which is a non-conservative field.
But how could there possibly be a faster and easier way to find an approximation of this non-conservative field without separately considering the electric field generated by the charges on the two plates of the capacitor?

 

[vanhees71]

You just use the usual quasistationary approximations of circuit theory and Faraday's Law then leads to
$$L \dot{i}=-CQ.$$
Using $\dot{Q}=i$, you get
$$L \ddot{Q}=-C Q \; \Rightarrow \; \ddot{Q}=-\omega^2 Q, \quad \omega=1/\sqrt{LC}$$
The solution is
$$Q(t)=c_1 \cos(\omega t) + c_2 \sin(\omega t),$$
with the integration constants $c_1$ and $c_2$ given by the initial condition, i.e., $Q(0)=Q_0$ and $i(0)=\dot{Q}(0)=i_0$.

 

[alan123hk]

This is just to find out the current, voltage and the amount of charge on the capacitor . I mean find out the electric field in all the surrounding space. The electric field in the entire space includes the induced electric field generated by the inductor and the electric field generated by the capacitor. They are stacked together.

I admit that there is only one non-conservative field, but on the other hand, how can there be a faster way to find this non-conservative field (even an approximation) directly without considering the electric fields generated by the inductor and capacitor separately?

 

[hutchphd]

This is just to find out the current, voltage and the amount of charge on the capacitor . I mean find out the electric field in all the surrounding space. The electric field in the entire space includes the induced electric field generated by the inductor and the electric field generated by the capacitor. They are stacked together.

I admit that there is only one non-conservative field, but on the other hand, how can there be a faster way to find this non-conservative field (even an approximation) directly without considering the electric fields generated by the inductor and capacitor separately?

In order to find a useful approximate solution, one must rigorously define the purpose and range of the approximation, and never lose sight of the validity thereof. Much good physics has been done by making assumptions and rigorously deriving the results thereof.
Hans Bethe, for instance, was famous for making the correct ansatz in an otherwise intractible situation. But he never lost sight of the limitations and sometimes figuring out why such an ansatz works better than expected reveals hidden details.

 

[SDL]

Can't we then try to describe and calculate the real electric field produced by a current-carrying conductor?

Maybe this reference would help:
https://web.mit.edu/6.013_book/www/chapter10/10.1.html
See "Example 10.1.2. Electric Field of a One-Turn Solenoid".

published in peer-reviewed format anywhere?

The reference above is a chapter from this book: Herman A. Haus, James R. Melcher. Electromagnetic Fields and Energy.
https://www.amazon.com/dp/013249020X/?tag=pfamazon01-20

 

[alan123hk]

Thank you for your replies, I think I need some time now to sort out and think about related issues.

 

[hutchphd]

Maybe this reference would help:
https://web.mit.edu/6.013_book/www/chapter10/10.1.html
See "Example 10.1.2. Electric Field of a One-Turn Solenoid".

Thanks. I don't believe I've seen that before. Very Nice.

 

[alan123hk]

Maybe this reference would help:
https://web.mit.edu/6.013_book/www/chapter10/10.1.html See "Example 10.1.2. Electric Field of a One-Turn Solenoid".

Thanks for the link, Example 10.12 demonstrates how to use good mathematical techniques to derive analytical solutions to physics problems.

Below I try to find potential and potential difference using the simple concept I've been talking about. It is to regard the entire cylindrical structure as a circuit to calculate. Of course, this does not have the ability to find the electric field distribution in the entire space, since Laplace's equation and other more advanced mathematics have to be used.

 

[hutchphd]

Of course, this does not have the ability to find the electric field distribution

Which is unique. QED.

 

[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. $$E= E_c+E ~~~~~\Rightarrow~~~~~E_c=0$$