Dielectric Constant of a Plasma

8/8/2019 Dielectric Constant of a Plasma

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http://farside.ph.utexas.edu/~rfitzp/teaching/em/lectures/node100.html

Dielectric constant of a plasma

A plasma is very similar to a gaseous medium, expect that the electrons arefree: i.e.,

there is no restoring force due to nearby atomic nuclii. Hence, we can obtain an

expression for the dielectric constant of a plasma from Eq. (1149) by setting to

zero, and to the number density of electrons, . We obtain

(1150)

where the characteristic frequency

(1151)

is called theplasma frequency. We can immediately see that formula (1150) is

problematic. For frequencies above the plasma frequency, the dielectric constant of

a plasma is less than unity. Hence, the refractive index is also less than unity.

This would seem to imply that high frequency electromagnetic waves can propagate

through a plasma with a velocity which is greaterthan the velocity of light in a

vacuum. Does this violate the principles of relativity? On the other hand, for

frequencies below the plasma frequency, the dielectric constant is negative, which

would seem to imply that the refractive index is imaginary. How should we

interpret this?

Consider an infinite plane-wave, of frequency, , greater than the plasma frequency,

propagating through a plasma. Suppose that the wave electric field takes the form


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(1152)

where it is understood that the physical electric field is the real partof the above

expression. A peak or trough of the above wave travels at the so-calledphase

velocity, which is given by

(1153)

Now, we have also seen that the phase velocity of electromagnetic waves in a

dielectric medium is , so

(1154)

It follows from Eq. (1150) that

(1155)

in a plasma. The above type of expression, which effectively determines the wave

frequency, , as a function of the wave-number, , for the medium in question, is

called a dispersion relation (since, amongst other things, it determines how fast

wave-pulses disperse in the medium). According to the above dispersion relation,

the phase velocity of high frequency waves propagating through a plasma is given by


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(1156)

which is indeed greater than . However, the theory of relativity does not forbid

this. What the theory of relativity says is that information cannot travel at a velocity

greater than . And the peaks and troughs of an infinite plane-wave, such as (1152),

do notcarry any information.

We now need to consider how we could transmit information through a plasma (orany other dielectric medium) by means of electromagnetic waves. The easiest way

would be to send a series of short discrete wave-pulses through the plasma, so that wecould encode information in a sort of Morse code.We can build up a wave-pulse from

a suitable superposition of infinite plane-waves of different frequencies and wave-

lengths: e.g.,

(1157)

where , and is determined from the dispersion relation

(1155). Now, it turns out that a relatively short wave-pulse can only be built up from

a superposition of plane-waves with a relatively wide range of different values.

Hence, for a short wave-pulse, the integrand in the above formula consists of the

product of a fairly slowly varying function, , and a rapidly oscillating function,

. The latter function is rapidly oscillating because the phase varies

very rapidly with , relative to . We expect the net result of integrating the

product of a slowly varying function and rapidly oscillating function to be small, since

the oscillations will generally average to zero. It follows that the integral (1157) is

dominated by those regions of -space for which varies least rapidlywith .


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Hence, the peak of the wave-pulse most likely corresponds to a maximum or

minimum of : i.e.,

(1158)

Thus, we infer that the velocity of the wave-pulse (which corresponds to the velocity

of the peak) is given by

(1159)

This velocity is called the group velocity, and is different to the phase velocity in

dispersive media, for which is not directly proportional to . (Of course, in a

vacuum, , and both the phase and group velocities are equal to .) The

upshot of the above discussion is that information (i.e., an individual wave-pulse)

travels through a dispersive media at the group velocity, rather than the phase

velocity. Hence, relativity demands that the group velocity, rather than the phase

velocity, must always be less than .

What is the group velocity for high frequency waves propagating through a plasma?

Well, differentiation of the dispersion relation (1155) yields

(1160)

Hence, it follows from Eq. (1156) that


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(1161)

which is less than . We thus conclude that the dispersion relation (1155) is indeed

consistent with relativity.

Let us now consider the propagation of low frequency electromagnetic waves througha plasma. We can see, from Eqs. (1156) and (1161), that when the wave frequency,

, falls below the plasma frequency, , both the phase and group velocities becomeimaginary. This indicates that the wave attenuates as it propagates. Consider, for

instance, a plane-wave of frequency . According to the dispersion relation

(1155), the associated wave-number is given by

(1162)

Hence, the wave electric field takes the form

(1163)

Indeed, it can be seen that for electromagnetic waves in a plasma take the

form of decaying standing waves, rather than traveling waves. We conclude that an

electromagnetic wave, of frequency less than the plasma frequency, which is

incident on a plasma will not propagate through the plasma. Instead, it will be totally

reflected.

We can be sure that the incident wave is reflected by the plasma, rather than absorbed,

by considering the energy flux of the wave in the plasma. It is easily demonstrated

that the energy flux of an electromagnetic wave can be written


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(1164)

For a wave with a real frequency and a complex -vector, the above formula

generalizes to

(1165)

However, according to Eq. (1162), the -vector for a low frequency electromagnetic

wave in a plasma ispurely imaginary. It follows that the associated energy flux is

zero. Hence, any low frequency wave which is incident on the plasma must be totally

reflected, since if there were any absorption of the wave energy then there would be

a net energy flux into the plasma.

The outermost layer of the Earth's atmosphere consists of a partially ionized zone

known as the ionosphere. The plasma frequency in the ionosphere is about 1MHz,

which lies at the upper end of the medium-wave band of radio frequencies. It followsthat low frequency radio signals (i.e., all signals in the long-wave band, and most in

the medium-wave band) are reflectedoff the ionosphere. For this reason, such signalscan be detected over the horizon. Indeed, long-wave radio signals reflect multiple

times off the ionosphere, with very little loss (they also reflect multiple times off the

Earth, which is enough of a conductor to act as a mirror for radio waves), and can

consequently be detected all over the world. On the other hand, high frequency radiosignals (i.e., all signals in the FM band) pass straight through the ionosphere. For this

reason, such signals cannot be detected over the horizon, which accounts for the

relatively local coverage of FM radio stations. Note, from Eq. (1151), that the plasma

frequency is proportional to the square root of the number density of free electrons.Now, the level of ionization in the ionosphere is maintained by ultra-violet light from

the Sun (which effectively knocks electrons out of neutral atoms). Of course, there isno such light at night, and the number density of free electrons in the ionosphere

consequently drops as electrons and ions gradually recombine. It follows that the

plasma frequency in the ionosphere also drops at night, giving rise to a marked

deterioration in the reception of distant medium-wave radio stations.

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Dielectric Constant of a Plasma