- #1

avjt

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Consider a 'current element' of length 'dl' carrying a current of [itex]I_{0} cos ({\omega}t)[/itex]. I understand the computation of the electric and magnetic fields due to this current element at any point [itex](r, \theta, \phi)[/itex], the computation of the Poynting vector at that point, and how the radiation resistance [itex]R_{r}[/itex] is derived from the

__time-averaged__Poynting vector integrated over the surface of an enclosing sphere.

Now, I am trying to derive the

__instantaneous power__passing through the surface of a sphere or radius 'r' centered at the current element (by integrating the Poynting vector directly over the surface of that sphere) and take the limit, at [itex]r\rightarrow\infty[/itex], so as to eliminate the effect of the near field entirely.

Given that the current is [itex]I_{0} cos ({\omega}t)[/itex], I was expecting to get a result of [itex]I_{0}^2R_{r} cos^2({\omega}t)[/itex]. But I keep getting [itex]I_{0}^2R_{r} sin^2({\omega}t)[/itex], or [itex]I_{0}^2R_{r} cos^2({\omega}t+{\pi}/2)[/itex] if you like.

Am I making a mistake (if so, can anyone point me to the correct derivation)? Or is there really a 90-degree phase shift between the current and the instantaneous power (if so, can anyone suggest some kind of physical explanation)?

Can anyone help me out? Thanks...

Avijit