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A capacitor has circular plates with radius $R$ and is being charged by a constant current $I$. The electric field $E$ between the plates is increasing, so the energy density is also increasing. This implies that there must be a flow of energy into the capacitor. Calculate the Poynting vector at radius r inside the capacitor (in terms of $r$ and $E$), and verify that its flux equals the rate of change of the energy stored in the region bounded by radius $r$.

Hi everybody, I don't want the answer, because actually I already know what it is, but I would really appreciate that someone answer my doubt:

While trying to answer this question, I was tentative to use the Maxwell equation $$\operatorname{curl} E = \frac{\partial B}{\partial t}$$ I would try to use this fact, and the fact that the electric field in this case is $$E = \frac{q}{A\epsilon}$$ So basically I would take the curl of this and put it in the Maxwell Equation I cited. But, I am not sure if we can do that! I would go on the math but I hesitated and tried to figure out if this is right (In my head, this electric field I calculated is for static fields, but, I do not see any equation that say this is wrong, so I am stuck here). What do you think?

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Let me give you two hints:

  1. Use the Ampère-Maxwell law in the global form( using Gauss theorem) $$ \oint_{\gamma} \vec{B} \cdot d \vec{l}=\frac{d }{dt}\Phi(\vec{E}) $$
  2. Use the cylindrical symmetry of the problem

N.B your doubt about the electric field is solved in the quasi-static approximation in which you consider only the first order of the electric field

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