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I am making som calculations on induction heating of a metallic cylinder inside

a solenoid through which an AC current passes, but the answer I get seems to

be completely unrealistic (10 kW with a 50 Hz, 5 Amps, 230 V rms), but I can't seem

to find my mistake.

What I think is wrong is the expression for the magnetic field inside the solenoid. I

have used

[tex] B = \mu n i [/tex]

But is this really correct for a time varying field?

As far as I know it is derived using amperes law in the static case when the displacement

current is zero, but I haven't found any passing comment or remark or anything anywhere

that this equation is valid only in the static case. So is this formula correct even in the

time varying (sinusoidal) case? If not, what should it be replaced by?

Basically, what I did for the rest of the calculation was simply to get the E-field by starting

from the maxwell equation

[tex] \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}[/tex]

and integrating both sides over a circle of radius r. The righthand side was simplified using the expression

for the B-field above, and the left hand side was simplified using Stoke's Theorem to obtain

an expression for the E-field at radius r from the cylinder axis.

Then I calculated the power from

[tex] P = \iiint_{\text{cylinder}}{\sigma E^2 \text{dV}} [/tex]

Another thing that worries me is that I find that the E-field is proportional to the distance

r to the cylinder axis. When I read about induction heating on the internet the skin effect is

always mentioned, but it didn't pop out my calculations anywhere... But on the other hand,

these calculations came straight from Maxwell's equations so I find it hard to believe that they are completely wrong...