20160127_PlasmaWeek1HW.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Theory\n",
    "\n",
    "**Warning** $T_e$ stands for $k_B T_e$!\n",
    "\n",
    "Debye length:\n",
    "\n",
    "$$\n",
    "\\lambda_D = \\sqrt{\\frac{\\epsilon_0 T_e}{e^2 n}}\n",
    "$$\n",
    "\n",
    "How to remember the formula:\n",
    "\n",
    "- Debye length: distance at which the Coulomb potential is screened by the plasma effects (collective effects)\n",
    "- if I heat the electrons, the Debye length gets bigger (their radius of movement increases)\n",
    "- if I make the plasma denser, the interaction length gets smaller (they are closer together)\n",
    "\n",
    "Electron plasma frequency:\n",
    "\n",
    "$$\n",
    "w_p = \\sqrt{\\frac{n e^2}{\\epsilon_0 m_e}}\n",
    "$$\n",
    "\n",
    "How to remember the formula:\n",
    "\n",
    "- Electron plasma frequency: the frequency with which the electrons naturally oscillates if I displace them (model: ions are fixed).\n",
    "- if I increase the density of the plasma, the mean distance between particles decreases: the oscillations become faster!\n",
    "- mass of electrons is at numerator: just like $\\sqrt{\\frac{k}{m}}$!\n",
    "- $\\epsilon_0 / e^2$ term just like in Debye length"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constants "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1eV = 1.602176565(35)×10−19 J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "kB×T[Kelvin]=T[Joules]=1.602×10−19×T[eV],"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "e = 1.602e-19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "epsilon_0 = 8.8542e-12 #F/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "m_e = 9.1094e-31 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "k_b = 1.3807e-23"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solar wind "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.4343569272229804"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = 10 * e\n",
    "n = 1e7\n",
    "debye = np.sqrt(epsilon_0 * T / (e**2 * n))\n",
    "debye"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "178378.68647273269"
      ]
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "wp = np.sqrt(n * e**2 / (epsilon_0 * m_e))\n",
    "wp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "28389.849694374814"
      ]
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     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "wp / (2 * np.pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "28 kHz = fréquence d'oscillation des électrons du soleil"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4108944014.0460491"
      ]
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     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nd = n * debye**3\n",
    "Nd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'4.1e+09'"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"{:.1e}\".format(Nd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lightning "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = 10 * e\n",
    "n = 1e23"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.4343569272229817e-08"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "debye = np.sqrt(epsilon_0 * T / (e**2 * n))\n",
    "debye"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17837868647273.266"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wp = np.sqrt(n * e**2 / (epsilon_0 * m_e))\n",
    "wp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'2.8e+12'"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"{:.1e}\".format(wp / (2 * np.pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.8 THz dans un éclair!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41.089440140460518"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nd = n * debye**3\n",
    "Nd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'4.1e+01'"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"{:.1e}\".format(Nd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'In the lightning setting, the debye length is 7.43e-08 m, the electron oscillation pulsation is 1.78e+13, or a frequency of 2.84e+12 Hz and the number of particles in a debye cube is 4.11e+01'"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"In the {} setting, the debye length is {:.2e} m, the electron oscillation pulsation is {:.2e}, or a frequency of {:.2e} Hz and the number of particles in a debye cube is {:.2e}\".format(\"lightning\", debye, wp, wp/(2*np.pi), Nd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Magnetic fusion plasma "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = 1e4 * e\n",
    "n = 1e20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.4343569272229809e-05"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "debye = np.sqrt(epsilon_0 * T / (e**2 * n))\n",
    "debye"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "564082935282.9021"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wp = np.sqrt(n * e**2 / (epsilon_0 * m_e))\n",
    "wp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'9.0e+10'"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"{:.1e}\".format(wp / (2 * np.pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41089440.140460506"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nd = n * debye**3\n",
    "Nd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'In the magnetic fusion setting, the debye length is 7.43e-05 m, the electron oscillation pulsation is 5.64e+11, or a frequency of 8.98e+10 Hz and the number of particles in a debye cube is 4.11e+07'"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"In the {} setting, the debye length is {:.2e} m, the electron oscillation pulsation is {:.2e}, or a frequency of {:.2e} Hz and the number of particles in a debye cube is {:.2e}\".format(\"magnetic fusion\", debye, wp, wp/(2*np.pi), Nd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# FLUORESCENT LAMP PLASMA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = k_b * 10000\n",
    "n = 1e17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8618601747815231"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T / e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.1825380030026764e-05"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "debye = np.sqrt(epsilon_0 * T / (e**2 * n))\n",
    "debye"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17837868647.27327"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wp = np.sqrt(n * e**2 / (epsilon_0 * m_e))\n",
    "wp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'2.8e+09'"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"{:.1e}\".format(wp / (2 * np.pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1039.645895990164"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nd = n * debye**3\n",
    "Nd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'In the magnetic fusion setting, the debye length is 2.18e-05 m, the electron oscillation pulsation is 1.78e+10, or a frequency of 2.84e+09 Hz and the number of particles in a debye cube is 1.04e+03'"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"In the {} setting, the debye length is {:.2e} m, the electron oscillation pulsation is {:.2e}, or a frequency of {:.2e} Hz and the number of particles in a debye cube is {:.2e}\".format(\"magnetic fusion\", debye, wp, wp/(2*np.pi), Nd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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