2005 II 1
From QED
The energy density in the plasma is:
So the total energy in the plasma is:
The ideal MHD equations with p = 0 give:
Combining:
So:
Plugging in:
Rearranging:
Or, using vector identities 1, 7 and 9:
Then:
Or:
Expanding the double cross product:
on the boundary:
The vacuum energy density is:
So:
Plugging in for :
Using the vector identity:
Since in the vacuum, and plugging in :
Expanding the double cross product, and changing the integral to a surface integral:
Since is zero on the surface:
There will be no jump conditions at the plasma-vacuum interface, since there are no surface currents. The time derivative of the total energy will then be:
The vacuum boundary contains the plasma-vacuum boundary and the vacuum-container boundary:
Where we have used the negative sign since the normal faces out of the vacuum, and so into the plasma. The total energy time derivative is then:
That is, the integral at the conducting wall of this quantity.
In MHD the E2 / 8π is considered small compared to the B2 / 8π term. Using Ohm's law:
So:
Then the electric field term over the magnetic field term gives:
So since .
This page was recovered in October 2009 from the Plasmagicians page on Generals_2005_II_1 dated 20:25, 15 May 2007.