Current is not continuous in RC circuits?

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Delta2
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yes that refers to X-rays, X-rays were discovered around 1895,Maxwell theoretically introduced the displacement current around 1860.
 
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vanhees71
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Displacement current was introduced to explain magnetic field induced by current. But how to understand time varying electric flux is like a current? I can't think of this invention other than compensating a missing term in Ampere's law.
That was the ingenious way Maxwell discovered the necessity for this term in his equations. It's, however, not a good interpretation in view of modern developments, particularly relativity. If you want to reveal the meaning of Maxwell's equations in a modern physical way, you should not consider this term as part of the current, but you should put it to the left side of the inhomogeneous Maxwell equation, so that the "true source", which is the current-density, is on the right-hand side, i.e., the Ampere-Maxwell Law should read (in Heaviside-Lorentz units, which are the most lucid choice to discuss fundamental relativistic properties of E&M)
$$\vec{\nabla} \times \vec{B} -\frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}.$$
If you interpret Maxwell's equations in this way, you are automatically lead to the physical solutions in the sense of "cause and effect", i.e., Jefimenko's retarded solutions (or equivalently the retarded potentials in Lorenz gauge): The causal sources of the em. field are the charge density and the current density and not some parts of the electromagnetic field.
 
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vanhees71
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There was Rontgen's experiment finding displacement current, why don't textbooks put Ampere's law and Rontgen's experiment/find on the same level as 2 parallel terms in the Maxwell's Eq involving curlB?
I'm not sure what you are referring to, but I guess it's about the Röntgen-Eichenwald experiment, conducted by Röntgen in 1888 and refined later by Eichenwald in 1903. This answered the question, whether the surface-charge density at the edge of a dielectricum in an electric field can induce a magnetic field when set into motion. Röntgen's and Eichenwald's experiment showed that this is indeed the case. One must know that this was not so clear in these days, because the microscopic theory about the nature of matter as consisting of charged particles was in its infancy, and the question about electrodynamics in moving bodies was an enigma, which was finally only resolved with the discovery of special relativity some years later. Today it's clear that any kind of current density, i.e., moving charges, lead to a magnetic field, but the Röntgen current is now clearly seen as the convection current of a surface-charge distribution, it's clear that it is a true surface-current density and thus not something like the "displacement current".

You find a very good discussion on early experiments, among them the Röntgen-Eichenwald experiments, about electrodynamics in moving media in

A. Sommerfeld, Lectures on Theoretical Physics, vol. 3

Despite the sin to use the ##\mathrm{i} c t## convention for the pseudometric, it contains a very good treatment of relativistic electrodynamics.

One should note that Sommerfeld was among the rare physicists who got the radiation field of a particle in hyperbolic motion right, which even Pauli had wrong in his famous review article about relativity in 1921!
 
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vanhees71
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These vacuum capacitors are, unfortunately, of low capacitance, so it's not easy to measure. The capacitor will charge quickly and then the current stops. But one basic set up is a battery, a high value resistor, and the capacitor all in series. Then you can monitor the voltage drop across the resistor, which is proportional to the capacitor current. You will probably need to borrow an oscilloscope or some other fast measuring device to see it. You also need to start your measurement with the capacitor voltage not equal to the battery voltage; for example, reverse the polarity of the battery somehow, or short out the capacitor first.
Well, if you connect a battery to an uncharged capacitor (taking into account the finite resistance of the wires, because otherwise it's a singualr problem) you have to solve the EoM:
$$Q/C+R \dot{Q}=U_0$$
with the solution
$$Q(t)=C U_0 [1-\exp(-t/(RC))].$$
The current is
$$i=\dot{Q}=\frac{U_0}{R} \exp(-t/(RC)).$$
 

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