Waves & Vibrations To turn a gas into a \"plasma,\" one or more electrons must b

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Waves & Vibrations

To turn a gas into a "plasma," one or more electrons must be completely separated from each nucleus. This can be accomplished by applying a very strong electric field (as in a spark or a lightning bolt), by the absorption of ultraviolet light (as happens in the "ionosphere," one of the upper layers of our atmosphere), or by heating to a very high temperature (as occurs in the sun). We will model a plasma as a "gas" of electrons with number density n. (In other words, there are n electrons per cubic meter.) Each electron has charge -e and mass m_e. Occupying the same volume is a gas of positively charged ions, each with charge +e. Because the ions were created by removing the electrons from neutral atoms, the number density of ions is also n. (In other words, there are n ions per cubic meter.) Consider a cube of plasma, with sidelength l. Somehow (e.g., by applying an electric field), the electrons are all displaced a small distance x to the right. This creates a layer of negative charge on the right, and leaves behind a layer of positive charge on the left, as shown in figure 1.P.1. For simplicity, treat each of these layers as an infinite thin sheet of charge. The layers create an electric field throughout the plasma, pointing to the right. This field exerts a force to the left on the electrons filling the central region of the figure, and to the right on the ions. However, we'll focus on the electrons because they are so much lighter, and assume that the ions remain stationary. When we turn off the original force that caused the displacement x of the cube of electrons, the electric field from the surface charge layers causes the electrons to spring back toward the left. However, because of their finite mass, they overshoot the equilibrium position (corresponding to displacement x = 0), move to the left of the ions, then are pulled back to the right, and so on, in an oscillatory motion. In this problem, you'll calculate the frequency of this oscillation. (a) Explain why the field produced by the combination of the two charge layers in the region between them is E = nex/epsilon_0. (You may need to refer back to your intro. E&M; textbook. Remember that we're treating the layers of charge as infinite sheets.) (b) Explain why this leads to a restoring force on the electrons F = -n^2 e^2 l^3/ EPSILON_0 x. (Remember that x

Explanation / Answer

a. electric field due to an infinitely large charged plate = sigma/2*epsilon [ where epsilon is permittivity of free space]

now assuming the right plate is filled with electrons, and the left with +vely charged particles, the two fields add up to give a field of E = sigma/epsilon inside

sigma, or charge density is charge/area

now number density of electrons = n

number of electrons = n*Area *x [ where x is the distance by which they were dispalced in the first palce]

so, charge = n*Area*x*e [ e is charge on electron]

so, E = n*e*x/epsilon

b. so, restoring force on electrons, F = charge on electrons * E

charge on electrons = e*Numebr of total electrons

q = e*n*l^2 [ where l^3 is volume of the total arrangement]

so, F = -n^2*e^2*l^3*x/epsilon [ - sign as electrons have -ve charge]

c. Now, for osscilation frequency

from newtons second law

F = ma

so ma = -n^2*e^2*l^3*x/epsilon [ where m is mass of all el;ectrons]

so |a/x| = n^2*e^2*l^3*/m*epsilon

w = sqroot(|a/x|) = sqroot(n^2*e^2*l^3*/m*epsilon)

now m = me*n*l^3 [ total mass = mass of electron*number density of electrons*volume]

so w = sqroot(n^2*e^2*l^3*/m*epsilon) = sqroot(n^2*e^2*l^3*/me*n*l^3*epsilon) = sqroot(n*e^2/me*epsilon)

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