lecture3: plasma physics 1 - columbia...
TRANSCRIPT
Homework #1 (1 of 3)
Problems 43
• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)
Γ = q2
4πε0a2W S kBT
.
Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.
Problems
2.1 Prove that the electron Debye length can be written as
λDe = 69 m!
T (K)
ne(m−3)
"1/2
2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).
2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?
2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).
2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.
2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from
1T
= ∂S∂λ
∂λ
∂U
and use#
ni = 1.
2
Homework #1 (2 of 3)
Problems 43
• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)
Γ = q2
4πε0a2W S kBT
.
Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.
Problems
2.1 Prove that the electron Debye length can be written as
λDe = 69 m!
T (K)
ne(m−3)
"1/2
2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).
2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?
2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).
2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.
2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from
1T
= ∂S∂λ
∂λ
∂U
and use#
ni = 1.3
Homework #1 (3 of 3)
Problems 43
• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)
Γ = q2
4πε0a2W S kBT
.
Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.
Problems
2.1 Prove that the electron Debye length can be written as
λDe = 69 m!
T (K)
ne(m−3)
"1/2
2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).
2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?
2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).
2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.
2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from
1T
= ∂S∂λ
∂λ
∂U
and use#
ni = 1.
4
Charged particle motion in inhomogeneous, static and slowly-varying electric and magnetic fields
Part 1: Gyromotion in Uniform Field
48 3 Single Particle Motion in Electric and Magnetic Fields
The magnetic induction B(r) of a dipole field is given by the expression
B(r) = µ0
4π3r(r · M) − r2M
r5 . (3.6)
Here r is a vector pointing from the magnetic dipole to the field point. At the author’slocation (54.3◦ N.; 10.1◦ E), the Earth magnetic field has a horizontal componentBh = 17, 700 nT and a vertical component Bv = −46, 150 nT.
3.1.4 E×B Drift
When we now allow for a stationary and homogeneous electric field, we can choosethe orientation of our coordinate system, without loss of generality, to have electricand magnetic field in the x-y plane, E = (Ex , 0, Ez) and B = (0, 0, Bz). In thiscase, Newton’s equation of motion reads
vx = qm
!Ex + vy Bz
"
vy = qm
(−vx Bz)
vz = qm
Ez . (3.7)
The motion along the magnetic field is now accelerated but independent of themotion in the x-y plane. According to the principle of superposition of motions,we can consider both effects separately. For the x-y plane, the motion can again bedecomposed
vx = −ω2cvx
vy = −ω2c (vy + Ex/Bz) . (3.8)
Again, we find a harmonic oscillation in x-direction, but the motion in y-directionis more complex. In a moving frame of reference, vy = vy + Ex/Bz , which movesat a constant velocity −Ex/Bz in negative y-direction, we obtain a simple harmonicmotion
¨vy = −ω2c vy . (3.9)
Thus the solution for the velocities is the superposition of a circular orbit and aconstant motion in the same plane. This constant motion is called the E×B-drift.For a particle that is initially at rest, the solution reads
vx = Ex
Bzsinωct
vy = Ex
Bz[cosωct − 1] . (3.10)
48 3 Single Particle Motion in Electric and Magnetic Fields
The magnetic induction B(r) of a dipole field is given by the expression
B(r) = µ0
4π3r(r · M) − r2M
r5 . (3.6)
Here r is a vector pointing from the magnetic dipole to the field point. At the author’slocation (54.3◦ N.; 10.1◦ E), the Earth magnetic field has a horizontal componentBh = 17, 700 nT and a vertical component Bv = −46, 150 nT.
3.1.4 E×B Drift
When we now allow for a stationary and homogeneous electric field, we can choosethe orientation of our coordinate system, without loss of generality, to have electricand magnetic field in the x-y plane, E = (Ex , 0, Ez) and B = (0, 0, Bz). In thiscase, Newton’s equation of motion reads
vx = qm
!Ex + vy Bz
"
vy = qm
(−vx Bz)
vz = qm
Ez . (3.7)
The motion along the magnetic field is now accelerated but independent of themotion in the x-y plane. According to the principle of superposition of motions,we can consider both effects separately. For the x-y plane, the motion can again bedecomposed
vx = −ω2cvx
vy = −ω2c (vy + Ex/Bz) . (3.8)
Again, we find a harmonic oscillation in x-direction, but the motion in y-directionis more complex. In a moving frame of reference, vy = vy + Ex/Bz , which movesat a constant velocity −Ex/Bz in negative y-direction, we obtain a simple harmonicmotion
¨vy = −ω2c vy . (3.9)
Thus the solution for the velocities is the superposition of a circular orbit and aconstant motion in the same plane. This constant motion is called the E×B-drift.For a particle that is initially at rest, the solution reads
vx = Ex
Bzsinωct
vy = Ex
Bz[cosωct − 1] . (3.10)
What drifts result from a static force that has the same sign for ions and electrons (like gravity)?
6
Key Dynamics
• Separate descriptions of perpendicular and parallel motion
• Fast gyration around B
• Slow perpendicular drift of gyro center
7
Guiding Center Motion in a Strong Magnetic Field
Part 2: Guiding Center Motion
Reference: I.B. Bernstein, “The Motion of a Charged Particle in a Strong Magnetic Field,” Advances in Plasma Physics 4,
311 (1971).
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Some Definitions & Cases
• 3 equations, 2nd order, 6 initial conditions
• Constant E and B
• Slowly varying E (with uniform B)
• Let E → 0, with B slowly varying in space
• …
Use Bernstein’s notation…
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