physics 122 Lecture 02 October 1st 2013_final

physics 122 Lecture 02 October 1st 2013_final - Lecture 2...

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Lecture 2 October 1, 2013
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COLLECTIVE BEHAVIOR By collec;ve behavior we mean mo;on that depends not only on local condi;ons but on the state of the plasma in remote regions as well Local concentra;ons of +ve and-­‐ve charges give rise to E fields. Mo;on of charges give currents and hence magne;c fields. These fields affect the mo;on of the plasma far away Compare with molecules of air no EM gravita;onal forces are negligible just collisions
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Concept of temperature Ο Π Ξ Μ Ν Λ = kT mu A u f 2 2 1 exp ) ( Figure I.4 where f is the number of particles per cm 3 with velocity between u and u + du . 2 2 1 mu is the kinetic energy k is the Boltzmann’s constant k = 1.38 × 10 -23 J/ o K The number n , or number of particles per cm 3 , is given by = . ) ( du u f n The constant A is related to the density n by 2 / 1 2 Ο Π Ξ Μ Ν Λ π = kT m n A The width of the distribution is characterized by the constant T which we call the temperature. To see the exact meaning of T , we can compute the average kinetic energy of particles in this distribution.
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= du u f du u f mu E av ) ( ) ( 2 1 2 Defining 2 / 1 ) / 2 ( m kT v th = and , / th v u y = we can express f ( u ) and E av as = = dy y Av dy y y mAv E v u A u f th th av th ) exp( ) exp( 2 1 ) / exp( ) ( 2 2 2 3 2 2 The integral in the numerator is integrable by parts [ ] [ ] + + + = = dy y dy y y y dy y y y ) exp( 2 1 ) exp( 2 1 ) exp( 2 1 ) exp( 2 2 2 2 Canceling the integrals we have kT mv Av mAv E th th th av 2 1 4 1 4 1 2 3 = = Ο Π Ξ Μ Ν Λ =
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Temperature (in 3-­‐D) E av =3KT/2 Since T and E av are so closely related, it is customary in plasma physics to give temperatures in units of energy. To avoid confusion on the number of dimensions involved it is not E av but rather the energy corresponding to kT that is used to denote the temperature. For kT = 1 eV = 1.6 × 10 -­‐19 J, we have Thus the conversion factor is 1 eV = 11,600 o K By a 2-­‐eV plasma we mean that kT = 2 eV, or E av = 3 eV in (3D) 600 , 11 10 38 . 1 10 6 . 1 23 19 = × × = T
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Debye Length DEBYE SHIELDING
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Debye Length ε 0 2 φ = ε 0 2 φ x 2 = e n i n e ( ) This comes from: ∇⋅ E = σ ε 0 E = −∇ φ ∇⋅ E = ∇⋅ −∇ φ ( ) = −∇ 2 φ
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f ( u ) = A exp 1 2 mu 2 + q φ " # $ % & ' kT e ( ) * + , - ( ) e e kT e n n φ exp = ε 0 d 2 φ dx 2 = en exp e φ kT e " # $ % & ' " # $ % & '− 1 ) * + + , - . . e x = 1 + x + x 2 2 ! + x 3 3 ! + ... x < 1 ε 0 d 2 φ dx 2 = en e φ kT e + 1 2 e φ kT e " # $ % & ' 2 + ... " # $ $ % & ' ' λ D = ε 0 kT e ne 2 !
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