Poynting's theorem in a solenoid

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The discussion revolves around verifying Poynting's theorem for a cylindrical solenoid connected to a resistance, with a current flowing through it. The user attempts to solve for the current using the inductance formula but struggles with expressing the current density and the complexity of calculations involved. Suggestions are made to consider alternative methods or resources to simplify the verification process. The conversation highlights the challenges of applying Poynting's theorem in this context and the need for a clearer understanding of current density in solenoids. Overall, the focus is on finding a more efficient approach to demonstrate the theorem's validity for the given scenario.
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Homework Statement


Consider a cylindrical solenoid(of radius a,and length h>>a,having n turns of wire for unity of lenght),The solenoid is connected to a resistance R and,at instant t=0 a current i(0)=i0 is flowing in the wire.Prove that poynting's theorem is verified for t>0[/B]

Homework Equations

[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/e6fe1dabe8c9699364d0e983f92139c7a6d4deeb[/B]

The Attempt at a Solution


I assumed the current i(t) could be obtained solving Ldi/dt=Ri,where L is the inductance I previously computed.I tried to verify the conservation of energy applying the theorem in its global form on the volume of the solenoid,but,on one side,i don't know how to express current density J for the solenoid,on the other i realized it would lead me to a massive amount of calculations,Maybe there's some other way I didn't take into account?Sorry for my English.[/B]
 
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