Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
volts per turn
contour
plasma
Figure 5.1. A subset of volts per turn coils on a contour l.
To interpret what we are measuring, notice that the poloidal (subscript p) and toroidal fields are
not coupled in Maxwell's equations, so that we can write Poynting's theorem for the poloidal
fields alone. To remind you, the basic equations needed are
∇ × B = µj 5.8
∇ × E = − ∂B
∂t
5.9
Multiplying these by -E and B/µ respectively, adding, and writing the poloidal component, gives
(φ is the direction the long way around the plasma)
∂
∂t
2
⎛ B
⎜ p ⎞ ⎛
⎟ + ∇ • ⎜ E × B p
⎝ 2µ 0
⎠ ⎝ µ 0
⎞
⎟ + j φ
E φ
= 0 5.10
⎠
Integrating over the volume V defined by rotating the contour l in the φ direction gives
∂
∂t
2
⎛ L i
I
⎜ p ⎞
⎝ 2 ⎠
⎟ + ∫ j E dV = ∫ εB dl
5.11
φ φ τ
V
l
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