By C.J. Joachain

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**Example text**

The electrons can respond to the variations in electric potential by moving along the field lines, so they remain in thermodynamic equilibrium along the magnetic field, n e ∝ exp(e ˜ /T ), or n˜ e e˜ ≈ , n T (59) with n˜ e the perturbation in the electron density and ˜ the perturbation in the electric potential. The ions move so slowly compared to the wave that the variation in the potential along the magnetic field has no direct effect on their motion. Variations in the ion density n˜ i are given by ∂ n˜ i /∂t + ∇ · (n v˜ E ) = 0 with v˜ E ≡ (−∇ ˜ ) × B/B 2 .

1) is −2m × Bex with m ≡ 12 (x × jpl ) d 3 x, the dipole moment of the plasma. If the aspect ratio R/a is large with (R, ϕ, Z ) cylindrical coordinates and a the minor radius of the plasma, m ≈ zˆ I π R 2 + ϕˆ {Bϕ2 − (Bϕ(ex) )2 } d 3 x/(4Bϕ(ex) ). Let Bp ≡ µ0 I /(2πa), then Bex · zˆ a =− Bp 4R 3p + Bpl2 2µ0 + Bp2 2µ0 Bϕ2 −(Bϕ(ex) )2 2µ0 d3x d3x , (33) with d 3 x = (πa 2 )(2π R) the volume of the plasma. If the externally produced vertical field, Bex · zˆ , becomes larger in magnitude than Bp , the poloidal magnetic field has a point of zero magnitude on the inboard side of the plasma.

If the magnetic field strength is toroidally symmetric, ∂ B/∂ϕ = 0, the canonical momentum pϕ is conserved. If the field is time independent, the Hamiltonian H is also conserved and the shape of the drift trajectories can be analytically determined. For trapped particles these trajectories have a characteristic banana shape in magnetic coordinates. If the field strength has no symmetries, the trapped particles are very sensitive to the exact form of the field P1: GNH/GJP P2: GSS Final pages Encyclopedia of Physical Science and Technology EN012B-582 July 26, 2001 10:27 389 Plasma Confinement strength.