1. Physics of Fusion power Lecture4 : Quasi-neutrality Force on the plasma

2. Quasi-neutrality Using the Poisson equation And a Boltzmann relation for the densities One arrives at an equation for the potential Positive added charge Response of the plasma

3. Solution The solution of the Poisson equation is Potential in vacuum Shielding due to the charge screening Vacuum and plasma solutionThe length scale for shielding is the Debye length which depends on both Temperature as well as density. It is around 10 -5 m for a fusion plasma

4. Quasi-neutrality For length scales larger than the Debye length the charge separation is close to zero. One can use the approximation of quasi-neutrality Note that this does not mean that there is no electric field in the plasma Under the quasi-neutrality approximation the Poisson equation can no longer be used to calculate the electric field

5. Divergence free current Using the continuity of charge Where J is the current density One directly obtains that the current density must be divergence free

6. Also the displacement current must be neglected From the Maxwell equation Taking the divergence and using that the current is divergence free one obtains The displacement current must therefore be neglected, and the relevant equation is

7. Quasi-neutrality The charge density is defined to be equal to zero (but a finite electric field does exist) This equation replaces the Poisson equation. (we do not calculate the electric field from Poisson’s equation, which would give zero field) Additionally, the displacement current is neglected. Length scales of the phenomena are larger than the Debye length, time scales longer than the plasma frequency. The current is divergence free.

8. Force on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is

9. Force on the plasma For the electric field Define an average velocity Then for the magnetic field

10. Force on the plasma Averaged over all particles Now sum over all species The total force density therefore is

11. Force on the plasma This force contains only the electro-magnetic part. For a fluid with a finite temperature one has to add the pressure force

12. Reformulating the Lorentz force Using The force can be written as Then using the vector identity

13. Force on the plasma One obtains Important parameter (also efficiency parameter) the plasma-beta Magnetic field pressure Magnetic field tension

14. Magnetic equilibria For stationary magnetic confinement device force balance implies than inside the plasma: Unfortunately, an isolated region plasma cannot be confined by its own currents. Pressure balance requires external confining force

16. Flux conservation When trying to change the magnetic flux through a metal ring an electric field is generated (Faraday) which drives a current such that it tries to conserve the flux The current eventually decays due to the resistivity A perfect conductor, however, would conserve the magnetic flux

18. Flux conservation A plasma is like a metal (electrons are free) A hot plasma has a small resistivity As a first approximation it is perfectly conducting Flux is then conserved but the fluid can be moving Flux is transported along with the fluid