Let u be the flow velocity of motional electromotive field exerted in plasma with magnetic field density B. There fore the Lorenz force can be written as
Now let us define current density J, electrical conductivity σ and electron mobility μe using above mentioned parameters as follows
here we assume flow velocity u is much less than electron velocity ue .
Now using the equation 1 and 2 and taking force balance into consideration we can get the generalized ohm's law equation in terms of above mentioned parameters as follows
where first term of the equation 3 gives Faraday current and the second term describes the hall current of the system.
Here the Faraday current is the direct consequence of the net electric force in the plasma, while the Hall current and Hall electric field are a consequence of the electromotive force (equation.1) acting on the electrons. The Lorentz force JxB primarily induced on electrons acts in the direction of retarding the fluid flow because of coulomb force between the electron and the ions, and the collision coupling between the ion and gaseous particles. The pressure gradient dp/dx balances more or less with the Lorentz and frictional force along the direction of flow which is usually along x shown in the picture above.
Keeping u constant is must to keep the motional induction. Therefore the force balance for unit volume plasma can be written as
thus the power generation by the Lorentz force per unit volume is then -u(JxB). -(J·E) is extracted to the external load and (J·J/ σ) dissipated as the ohmic heat in the plasma. The electrical power balance can be obtained from equation (3) is
the rate of enthalpy 'h' drops per unit volume will approximately balances with the sum of J·E and radiation heat loss qloss. There fore the power balance for the plasma of unit volume is given by
where ρ is the gas density.
| Figure shows orientation of each vector fields. |
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