Thus B).u The second term of this expression is simply the rate at which the Lorentz force jx B does mechanical work on the plasma moving at u. The first term is always positive and is the familiar Joule heating(also called Ohmic heating) effect. Notice how the presence of the magnetic field introduces the possibility of accelerating a plasma, in addition to the unavoidable heating In an efficient accelerator, we would try to maximize(jxB). u at the expense of i/o Origin of the magnetic field steady state and without magnetic materials)is Ampere's law een b and j(inthe The magnetic field can either be provided by external coils, or induced directly by currents circulating in the plasma. The general relationship betwe where Ho =4X10(in MKS units)is the permeability of vacuum In integral form .A=手Bd (17) which states that the circulation of - around a closed line equals the total current linked by the loop. When the current is constrained to circulate in metallic wires, the integral form can be used to provide simple formulae for the field due to various conductor arrangements. For example, inside a long solenoid carrying a current I the field B is nearly constant and we obtain(see sketch 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 6 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 6 of 21 Thus ( ) 2 j W = + j × B . u σ G JG G (15) The second term of this expression is simply the rate at which the Lorentz force j × B G JG does mechanical work on the plasma moving at u G . The first term is always positive and is the familiar Joule heating (also called Ohmic heating) effect. Notice how the presence of the magnetic field introduces the possibility of accelerating a plasma, in addition to the unavoidable heating. In an efficient accelerator, we would try to maximize ( ) j × B . u G JG G at the expense of 2 j σ . Origin of the magnetic field The magnetic field can either be provided by external coils, or induced directly by the currents circulating in the plasma. The general relationship between B JG and j G (in steady state and without magnetic materials) is Ampere’s law 0 B j = × ∇ µ JG G (16) where -7 µ π 0 = 4 ×10 (in MKS units) is the permeability of vacuum. In integral form 0 B j . dA = .dl µ ∫∫ ∫ JG GG G v (17) which states that the circulation of 0 B µ JG around a closed line equals the total current linked by the loop. When the current is constrained to circulate in metallic wires, the integral form can be used to provide simple formulae for the field due to various conductor arrangements. For example, inside a long solenoid carrying a current I, the field B JG is nearly constant, and we obtain (see sketch)