Chapter 6   Electromagnetic Waves in Plasmas

6.1   Introduction

• Solar Atmosphere
• Interplanetary and Interstellar Space
• Fusion Devices, such as the Tokamak
• Earth's Ionosphere

Solids:

metals

Liquids:

conducting fluids (such as Mercury (Hg))

Gases:

this is the most common form of a plasma

• energetic photons
• particle-particle interaction

6.2   Conduction in Plasmas

• e>0 by definition
• t_collision is a typical timescale for the loss of mementum

• there is a steady magnetic field, so as the electon moves it experiences a force
• the ions are stationary on the timescale of the electromagnetic waves frequency due to their much bigger mass (about 2000 times bigger)

Case A:

t_collision<<1/w where the collisions dominate

jf=

æ ç ç è

Ne2tcollision me

ö ÷ ÷ ø

E=sE

where s is the same one used in Section 5

Case B:

t_collision>>1/w where the collisions are weak

jf=

Ne2 mew

E     (6.3)

in this case the s is represented by Ne²/m_ewE .

• s depends on 1/w
• j_f and E are out of phase by p/2 , they are out of phase. This occurs as in a collisionless material the electrons have to be slowed down before they turn around, this causes the current to go out of phase with the electric field
• the is no power dissipated for the electromagnetic wave in a collisionless plasma
  <j_f.E>=0  (Problem Sheet 7)

6.3   The Wave Equation in a Collisionless Plasma

w <w₀ :

the solution is a totally imaginary wave, this is known as an evanescant wave. So a wave with w <w₀ incident on a plasma is reflected. The magnitude of the evanescant wave is of the order c/w_p

w >>w₀ :

if w~ kc we get a pure electromagnetic wave solution

ww_p :

we get a dispersive wave, where the phase velocity is

vp=

w k

=c

æ ç ç ç ç ç è

1-

æ ç ç è

wp w

ö ÷ ÷ ø

2

ö ÷ ÷ ÷ ÷ ÷ ø

- 1 2

³ c     (6.8)

although the phase velocity is travelling faster than the speed of light the actual energy transmission is at the group velocity which is lower than the speed of light, so relativity is not violated

vg=

¶w ¶ k

=c

æ ç ç ç ç ç è

1-

æ ç ç è

wp w

ö ÷ ÷ ø

2

ö ÷ ÷ ÷ ÷ ÷ ø

1 2

£ c     (6.9)

Figure 6.1 - We Note That v_pv_g=c²

The Ionosphere:

N~ 10¹² m^-3

n_p~ 9 Mhz

Metals:

N~ 10²⁹ m^-3

n_p~ 3× 10¹⁵ Hz

6.4   Examples of Plasmas

6.4.1   The Earth's Ionosphere

Figure 6.2 - The Profile of the Earth's Ionsphere

Figure 6.3 - Reflection of Radio Waves Off The Ionosphere

6.4.2   Metals

• for when w <w_p (for copper this is about 3× 10¹⁵ Hz ) we get that the radiation is reflected (via total internal reflection)
  R~ 1
• for when w >w_p the metal becomes transparent to the radiation and so
  R® 0
  and
  n_metal® 1

X-Ray Telescope