Chapter 1, Introduction Video Solutions, Introduction to Plasma Physics and Controlled Fusion volume 1 plasma physics

Problem 1

Compute the density (in units of $\mathrm{m}^{-3}$ ) of an ideal gas under the following conditions;
(a) At $0^{\circ} \mathrm{C}$ and 760 Torr pressure (1 Torr $=1 \mathrm{~mm} \mathrm{Hg}$ ). This is called the Loschmidt number.
(b) In a vacuum of $10^{-3}$ Torr at room temperature $\left(20^{\circ} \mathrm{C}\right)$. This number is a useful one for the experimentalist to know by heart $\left(10^{-3}\right.$ Torr $=1$ micron).

Anand Jangid

Numerade Educator

01:00

Problem 2

Derive the constant $A$ for a normalized one-dimensional Maxwellian distribution
$$
\hat{f}(u)=A \exp \left(-m u^{2} / 2 K T\right)
$$
such that
$$
\int_{-\infty}^{\infty} \hat{f}(u) d u=1
$$

Dominador Tan

Numerade Educator

01:15

Problem 3

On a log-log plot of $n_{e}$ vs. $K T_{e}$ with $n_{e}$ from $10^{6}$ to $10^{25} \mathrm{~m}^{-3}$, and $K T_{e}$ from $0.01$ to $10^{5} \mathrm{eV}$, draw lines of constant $\lambda_{\mathrm{D}}$ and $N_{\mathrm{D}} .$ On this graph, place the following points $\left(n\right.$ in $\mathrm{m}^{-9}, K T$ in $\left.\mathrm{eV}\right)$ :
1. Typical fusion reactor: $n=10^{21}, K T=10,000$
2. Typical fusion experiments: $n=10^{19}, K T=100$ (torus); $n=10^{2.3}, K T=$ 1000 (pinch).
3. Typical ionosphere: $n=10^{11}, K T=0.05$.
4. Typical glow discharge: $n=10^{15}, K T=2$.
5. Typical flame: $n=10^{14}, K T=0.1 .$
6. Typical Cs plasma; $n=10^{17}, K T=0.2$.
7. Interplanetary space: $n=10^{6}, K T=0.01$.
Convince yourself that these are plasmas.

Katie Mcalpine

Numerade Educator

03:36

Problem 4

Compute the pressure, in atmospheres and in tons/ft ${ }^{2}$, exerted by a thermonuclear plasma on its container. Assume $K T_{e}=K T_{i}=20 \mathrm{keV}, n=10^{21} \mathrm{~m}^{-3}$, and $p=n K T$, where $T=T_{i}+T_{e}$

Anand Jangid

Numerade Educator

00:35

Problem 5

In a strictly steady state situation, both the ions and the electrons will follow the Boltzmann relation
$$
n_{i}=n_{0} \exp \left(-q_{i} \phi / K T_{j}\right)
$$
For the case of an infinite, transparent grid charged to a potential $\phi$, show that the shielding distance is then given approximately by
$$
\lambda_{\mathrm{D}}^{-2}=\frac{n e^{2}}{\epsilon_{0}}\left(\frac{1}{K T,}+\frac{1}{K T_{i}}\right)
$$
Show that $\lambda_{\mathrm{D}}$ is determined by the temperature of the colder species.

Salamat Ali

Numerade Educator

11:54

Problem 6

An alternative derivation of $\lambda_{\mathrm{D}}$ will give further insight to its meaning. Consider two infinite, parallel plates at $x=\pm d$, set at potential $\phi=0 .$ The space between them is uniformly filled by a gas of density $n$ of particles of charge $q$.
(a) Using Poisson's equation, show that the potential distribution between the plates is
$$
\phi=\frac{n q}{2 \epsilon_{0}}\left(d^{2}-x^{2}\right)
$$
(b) Show that for $d>\lambda_{\mathrm{D}}$, the energy needed to transport a particle from a plate to the midplane is greater than the average kinetic energy of the particles.

Ren Jie Tuieng

Numerade Educator

01:51

Problem 7

Compute $\lambda_{D}$ and $N_{\mathrm{D}}$ for the following cases:
(a) A glow discharge, with $n=10^{16} \mathrm{~m}^{-3}, K T_{t}=2 \mathrm{eV}$.
(b) The earth's ionosphere, with $n=10^{12} \mathrm{~m}^{-3}, K T_{e}=0.1 \mathrm{eV}$.
(c) A $\theta$-pinch, with $n=10^{24} \mathrm{~m}^{-3}, K T_{ }=800 \mathrm{eV}$.

Averell Hause

Carnegie Mellon University

01:24

Problem 8

In laser fusion, the core of a small pellet of DT is compressed to a density of $10^{33} \mathrm{~m}^{-3}$ at a temperature of $50,000,000^{\circ} \mathrm{K}$. Estimate the number of particles in a Debye sphere in this plasma.

Ze-Han Lee

Numerade Educator

02:18

Problem 9

A distant galaxy contains a cloud of protons and antiprotons, each with density $n=10^{6} \mathrm{~m}^{-3}$ and temperature $100^{\circ} \mathrm{K} .$ What is the Debye length?

Sanjeev Kumar

Numerade Educator

00:41

Problem 10

A spherical conductor of radius $a$ is immersed in a plasma and charged to a potential $\phi_{0} .$ The electrons remain Maxwellian and move to form a Debye shield, but the ions are stationary during the time frame of the experiment. Assuming $\phi_{0} \ll K T_{e} / e$, derive an expression for the potential as a function of $r$ in terms of $a, \phi_{0}$, and $\lambda_{\mathrm{D}}$ (Hint: Assume a solution of the form $e^{-k r} / r .$ )

Manish Kumar

Numerade Educator

03:09

Problem 11

A feld-effect transistor (FET) is basically an electron valve that operates on a finite-Debye-length effect. Conduction electrons flow from the source S to the drain D through a semiconducting material when a potential is applied between them. When a negative potential is applied to the insulated gate $G$, no current can flow through $\mathrm{G}$, but the applied potential leaks into the semiconductor and repels electrons. The channel width is narrowed and the electron flow impeded in proportion to the gate potential. If the thickness of the device is too large, Debye shielding prevents the gate voltage from penetrating far enough. Estimate the maximum thickness of the conduction layer of an $n$-channel FET if it has doping level (plasma density) of $10^{422} \mathrm{~m}^{-3}$, is at room temperature, and is to be no more than 10 Debye lengths thick. (See Fig. PI-11.)

Chai Santi

Numerade Educator