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Using Ampere we get:

[itex]H=In\hat{z}[/itex]

[itex]B=\mu In \hat{z}[/itex]

Using Faraday we get:

[itex]E=-0.5\mu n I_0 r \hat{\theta}[/itex]

the Poynting Vector is:

[itex]\vec{E} \times \vec{H}[/itex]

Integrating over the surface of some volume inside the solenoid to find the power flowing out, we get:

[itex]\int \vec{N}.\vec{dS} = -\pi \mu n^2 l (I_0)^2 r^2 t[/itex]

Also, the rate of change of energy stored in the magnetic field comes out as:

[itex]\pi \mu n^2 l (I_0)^2 r^2 t = \frac{dU}{dt}[/itex]

Also, work done against field (for that volume):

[itex] - \xi I = \pi \mu n^2 r^2 l (I_0)^2 t = \frac{dW}{dt} [/itex]

These three things don't seem to match up to the energy continuity equation - what am I thinking wrong?