lab7.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy.optimize import curve_fit\n",
    "from scipy.optimize import leastsq\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Temperature = 23 degrees C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polarization Properties of the Generator and Receiver\n",
    "With the transmitter and receiver facing each other along a straight goniometer track, rotate one horn\n",
    "about its horizontal axis in 5° increments, and record the signal. Given that the receiver detects only one\n",
    "polarization, how do you expect the detected signal to vary if the transmitted waves are linearly\n",
    "polarized? Make a plot of this data; comment on if this is consistent with the data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Call the angle relative to the upright position on this axis $\\phi$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We would expect the strength of the detected signal to vary sinusoidally if the transmitted waves are linearly polarized. Think of it like having a fixed vector which represents the polarization of a wave, then imagine another vector that can be rotated which represents the angle of the receiver. The projection of the first vector on the second depends on th cosine of the angle between them, which is the same as the sine with a phase shift."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "phis = np.arange(0,185,5)\n",
    "currents = [.50, .52, .50, .48, .48, .46, .42, .38, .36, .32, .28, .22, .18, .14, .10, .08, .02, .02, .00, .00, .02, .04, .06, .10, .16, .20, .24, .28, .32, .36, .40, .43, .46, .48, .50, .50, .52]\n",
    "polardata = pd.DataFrame({'phi':phis, 'current': currents}, columns=['phi', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs relative horn angle $\\\\phi$')"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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BX/nKV3Kvef7553Mb/5/97Gd88pOfZJdddmHVqlW5+evWrePRRx/tcVxZY+zK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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd385620828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(polardata['phi'], polardata['current'], label='experimental data')\n",
    "plt.xlabel(r'angle ($^\\circ$)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs relative horn angle $\\phi$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data does appear to behave as we predicted."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Spatial Dependence of Transmitter Signal\n",
    "Determine the dependence of the transmitter signal on the transmitter-receiver separation. Put the\n",
    "transmitter and receiver facing each other along a straight goniometer track, with the same polarization\n",
    "orientation for both. Start at a 45 cm separation and adjust the intensity/variable sensitivity dials on the\n",
    "receiver so that the meter reads 1.00. Then move the source and receiver farther apart, taking receiver\n",
    "readings at about 5 cm increments. From this data determine the r-dependence of the microwave\n",
    "receiver signal. Create a plot of this data and fit it to the curve $Ar^n$. Report the values $A \\pm \\delta A$ and $n \\pm \\delta n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [],
   "source": [
    "separations = np.arange(65, 115, 5)\n",
    "currents = [1.00, .84, .78, .76, .66, .58, .56, .52, .52, .48]\n",
    "spatialdata = pd.DataFrame({'separation':separations, 'current': currents}, columns=['separation', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(r, A, n):\n",
    "    return A*r**n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [],
   "source": [
    "params, cov = curve_fit(f, spatialdata['separation'], spatialdata['current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(r):\n",
    "    return params[0]*r**params[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [],
   "source": [
    "est = f([r for r in spatialdata['separation']])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [],
   "source": [
    "std = np.sqrt(np.diag(cov))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fitted equation is $344 r^{-1.4}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = 344 \\pm 114$\n",
    "\n",
    "$r = -1.4 \\pm .076$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs horn separation')"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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d0zBjBlx88XHvMvvQ2OTUNIbP/R7AEoMxIZRf6exHRSTXiX5F5FIRyfMmNlNK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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd3855c9f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(spatialdata['separation'], spatialdata['current'], label='experimental data')\n",
    "plt.plot(spatialdata['separation'], est, c=(1,0,0), label='fitted data')\n",
    "plt.xlabel('separation (cm)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title('current vs horn separation')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The strength of the detected signal is likely theoretically inversely proportional to the separation. Inconsistencies between this prediction and the experimentally determined $I_{detected} \\propto r^{-1.4}$ is likely due to inaccuracies in the experiment."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Angular Dependence of Transmitter Signal\n",
    "Place the transmitter directly over the protractor plate of the goniometer and position the receiver at\n",
    "the far end of the other arm. Rotate the receiver arm to study the angular dependence of the\n",
    "microwave signal radiated from the transmitter in 10 degree increments. What you want is the angular \n",
    "dependence of the microwave signal in the equatorial plane of the dipole source. Make sure you have\n",
    "both the transmitter and detector oriented with their polarizations vertical. Make a plot of this data.\n",
    "From the spatial and angular measurements, do you conclude that the waves are plane waves or\n",
    "spherical waves? (Even though the signal is not proportional to intensity, you can assume that they\n",
    "share the same r-dependence.) See the Appendix for a bit more on spherical and plane waves."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Call the angle relative to the upright position on this axis $\\theta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sep = 60\n",
    "thetas = np.arange(0, 100, 10)\n",
    "currents = [1.00, .68, .32, .12, .04, .00, .00, .00, .00, .00]\n",
    "angulardata = pd.DataFrame({'theta':thetas, 'current': currents}, columns=['theta', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs relative horn angle $\\\\theta$')"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd38593c0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(angulardata['theta'], angulardata['current'], label='experimental data')\n",
    "plt.xlabel(r'angle ($^\\circ$)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs relative horn angle $\\theta$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the intensity of the detected signal is inversely proportional to the separation, and planar waves have no such deterioration in intensity, the waves are not planar. Since the intensity also deteriorates with an increasing angle between the detector and the transmitter, the waves are also not spherical. (I think they're maybe cylindrical.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explore Polarization Phenomenon\n",
    "Start with the receiver and transmitter horns orthogonally oriented about their axes so the measured\n",
    "receiver signal is a minimum (“crossed” transmitter and receiver). Then place the slotted metal sheet\n",
    "polarizer in front of the transmitter horn and record the signal as you rotate the polarizer about the axis\n",
    "between transmitter and receiver in 10 degree increments. (One of the horns has a protractor scale to\n",
    "make this measurement easier.) Make a plot of the data and fit it to a sine or cosine function\n",
    "appropriately. Does the slotted sheet simply block polarization perpendicular to the slots, or is\n",
    "something more complicated happening? What physical model might explain the observations?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Call the angle relative to the upright position on this axis $\\psi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sep = 60\n",
    "psis = np.concatenate((np.arange(0, 50, 10), np.array([45]), np.arange(50, 70, 10)), axis=0)\n",
    "currents = [.00, .06, .10, .58, .80, .86, .90, .82] \n",
    "angulardata = pd.DataFrame({'psi':psis, 'current': currents}, columns=['psi', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fit_sin(tt, yy):\n",
    "    '''Fit sin to the input time sequence, and return fitting parameters \"amp\", \"omega\", \"phase\", \"offset\", \"freq\", \"period\" and \"fitfunc\"'''\n",
    "    tt = np.array(tt)\n",
    "    yy = np.array(yy)\n",
    "    ff = np.fft.fftfreq(len(tt), (tt[1]-tt[0]))   # assume uniform spacing\n",
    "    Fyy = abs(np.fft.fft(yy))\n",
    "    guess_freq = abs(ff[np.argmax(Fyy[1:])+1])   # excluding the zero frequency \"peak\", which is related to offset\n",
    "    guess_amp = np.std(yy) * 2.**0.5\n",
    "    guess_offset = np.mean(yy)\n",
    "    guess = np.array([guess_amp, 2.*np.pi*guess_freq, 0., guess_offset])\n",
    "\n",
    "    def sinfunc(t, A, w, p, c):  return A * np.sin(w*t + p) + c\n",
    "    popt, pcov = curve_fit(sinfunc, tt, yy, p0=guess)\n",
    "    A, w, p, c = popt\n",
    "    f = w/(2.*np.pi)\n",
    "    fitfunc = lambda t: A * np.sin(w*t + p) + c\n",
    "    return {\"amp\": A, \"omega\": w, \"phase\": p, \"offset\": c, \"freq\": f, \"period\": 1./f, \"fitfunc\": fitfunc, \"maxcov\": np.max(pcov), \"rawres\": (guess,popt,pcov)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [],
   "source": [
    "res = fit_sin(angulardata['psi'], angulardata['current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "est = res['fitfunc'](angulardata['psi'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [],
   "source": [
    "std = np.sqrt(np.diag(res['rawres'][2]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(\\psi) = A \\sin{(\\omega \\psi + \\phi)} + c$ (different $\\phi$ than previous problems)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fitted equation is $-0.467 \\sin{(0.070 \\psi + 1.172)} + 0.446$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = -0.467 \\pm .031$\n",
    "\n",
    "$\\omega = 0.070 \\pm .006$\n",
    "\n",
    "$\\phi = 1.172 \\pm .214$\n",
    "\n",
    "$c = 0.446 \\pm .028$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs relative horn angle $\\\\psi$')"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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FmcCXwHmquhZARF6OeFTGxJsXX4S33oL+/d0IqMZEqbySQhNcV9L/isga3FAV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\nPyEdBffT6F0QAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd3854ce438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(angulardata['psi'], angulardata['current'], label='experimental data')\n",
    "plt.plot(angulardata['psi'], est, c=(1,0,0), label='fitted data')\n",
    "plt.xlabel(r'angle ($^\\circ$)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs relative horn angle $\\psi$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We did this one in two ways, first by rotated the polarizer (above) and the second by rotating the transmitter (below)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sep = 60\n",
    "psis = np.concatenate((np.arange(0, 50, 10), np.array([45]), np.arange(50, 140, 10),np.array([135]), np.arange(140, 190, 10)), axis=0)\n",
    "currents = [.08, .08, .08, .06, .04, .03, .02, .02, .01, .00, .00, .00, .00, .02, .03, .04, .04, .06, .06, .08, .08]\n",
    "angulardata = pd.DataFrame({'psi':psis, 'current': currents}, columns=['psi', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "res = fit_sin(angulardata['psi'], angulardata['current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "est = res['fitfunc'](angulardata['psi'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "std = np.sqrt(np.diag(res['rawres'][2]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(\\psi) = A \\sin{(\\omega \\psi + \\phi)} + c$ (different $\\phi$ than previous problems)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fitted equation is $0.043 \\sin{(.033 \\psi + 1.761)} + 0.040$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = 0.043 \\pm .002$\n",
    "\n",
    "$\\omega = 0.033 \\pm .002$\n",
    "\n",
    "$\\phi = 1.761 \\pm .169$\n",
    "\n",
    "$c = .040 \\pm .003$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs relative horn angle $\\\\psi$')"
      ]
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Fza5n5aJNUWRnww03WPNTlAs3Wfwd+AioKyKvA58BQyMWlYkN550Hd90F//oX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\njDEmJEsWxhhjQvp/j7wmk0m8nmsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd38531a668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(angulardata['psi'], angulardata['current'], label='experimental data')\n",
    "plt.plot(angulardata['psi'], est, c=(1,0,0), label='fitted data')\n",
    "plt.xlabel(r'angle ($^\\circ$)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs relative horn angle $\\psi$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The detected intensity of the signal varies sinusoidally like vector projections, so it seems that instead of blocking waves that are perpendicular to it, the polarizer blocks the perpendicular **components** of all of the waves that pass through it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because the transmitted wave will have absolutely no component in the direction of the receiver's detection direction, the receiver simply will not detect anything. However, when the slotted metal plate was added, a signal was detected. According to Malus' law, \n",
    "$I=I_{max}\\cos^2{\\phi}$\n",
    "\n",
    "Given that the plate and transmitter were initially perpendicular, Malus' law states that the intensity measured would be zero at 90 degrees, which it was, and it increased to it's maximum when the transmitter was at 0 and 180 degrees and produced waves parallel to the slots in the plate, meaning that the plate did simply block polarization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explore Reflection Phenomena\n",
    "Verify that microwaves incident on a metal plate follow the law of specular reflection. With the\n",
    "transmitter at one end of one goniometer track, and the receiver at the other, set a fixed angle between\n",
    "the two tracks somewhere between 70 degrees and 110 degrees. Mount a solid metal plate on the\n",
    "goniometer protractor. Now record the detected signal while rotating the metal plate in 10 degree\n",
    "increments, keeping the angle between the transmitter and receiver fixed. Submit a plot of this\n",
    "experiment."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Data for this section was collected by Aidan McGirr's group as this section was accidentally ommitted on account of the lab manual change.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "angles = [90, 100, 110, 120, 130, 140, 150, 160, 170, 180]\n",
    "currents = [0, .01, .03, .04, .39, .2, .03, 0, 0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "reflectordata = pd.DataFrame({'angle':angles, 'current':currents}, columns=['angle', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs angle with reflector')"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HA0+l08XtG9Uece+LD+Bm4HlgM8mb/NPAUOBX6T/1V8AeaV0B15B8k2I50Fjt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\nmJlZxknBzMwyTgpmZpZxUjAzs8z/BwjbVyllKswMAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd38515d198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(reflectordata['angle'], reflectordata['current'])\n",
    "plt.xlabel(r'angle ($^\\circ$)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs angle with reflector')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use Standing Waves to Measure the Wave Speed\n",
    "Mount a metal plate on the goniometer's receiver arm to reflect the microwaves directly back into the\n",
    "transmitter. This can give rise to standing waves between the transmitter horn and the reflector plate.\n",
    "Detect them by mounting the detector probe between the transmitter and reflector and connecting its\n",
    "output to the receiver. The meter now reads the signal at the probe’s position. Adjust the positions of\n",
    "the transmitter and reflector to get a strong standing wave, and then map out the standing wave signal\n",
    "with the detector probe. Record the position(s) at which signal minima are observed, and use this to\n",
    "determine the microwave wavelength. Combine this result with the known generator frequency (10.5\n",
    "GHz) to determine the speed of microwave propagation.\n",
    "Make a plot of your data to visualize the standing wave and report the wave speed as 𝜈 ± 𝛿𝜈 with\n",
    "appropriate units. Does this number look familiar? For a perfect standing wave pattern, what should\n",
    "the wave amplitudes be at the reflector and transmitter ends (are they nodes or antinodes)?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Data for this section was collected by Madison Grayson's group as the standing wave generated by this group did not produce multiple extrema.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "positions = [30.5, 30.6, 30.7, 30.8, 30.9, 31, 31.1, 31.2, 31.3, 31.4, 31.5, 32, 32.5, 33, 33.5, 33.6, 33.7,\n",
    "             33.8, 33.9, 34, 34.5, 35, 35.1, 35.2, 35.3, 35.4, 35.5, 35.6, 35.7, 35.8, 35.9, 36, 36.5, 37]\n",
    "currents = [0.38, 0.42, 0.42, 0.42, 0.48, 0.43, 0.5, 0.56, 0.56, 0.53, 0.48, 0.46, 0.48, 0.4, 0.46, 0.48,\n",
    "            0.52, 0.53, 0.52, 0.46, 0.34, 0.34, 0.38, 0.39, 0.39, 0.36, 0.38, 0.36, 0.34, 0.3, 0.24, 0.28, 0.36, 0.43]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [],
   "source": [
    "standwaves = pd.DataFrame({'position':positions, 'current':currents}, columns=['position', 'current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>position</th>\n",
       "      <th>current</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>30.5</td>\n",
       "      <td>0.38</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>30.6</td>\n",
       "      <td>0.42</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>30.7</td>\n",
       "      <td>0.42</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>30.8</td>\n",
       "      <td>0.42</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>30.9</td>\n",
       "      <td>0.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>31.0</td>\n",
       "      <td>0.43</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>31.1</td>\n",
       "      <td>0.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>31.2</td>\n",
       "      <td>0.56</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>31.3</td>\n",
       "      <td>0.56</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>31.4</td>\n",
       "      <td>0.53</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>31.5</td>\n",
       "      <td>0.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>32.0</td>\n",
       "      <td>0.46</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>32.5</td>\n",
       "      <td>0.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>33.0</td>\n",
       "      <td>0.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>33.5</td>\n",
       "      <td>0.46</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>33.6</td>\n",
       "      <td>0.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>33.7</td>\n",
       "      <td>0.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>33.8</td>\n",
       "      <td>0.53</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>33.9</td>\n",
       "      <td>0.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>34.0</td>\n",
       "      <td>0.46</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>34.5</td>\n",
       "      <td>0.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>35.0</td>\n",
       "      <td>0.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>35.1</td>\n",
       "      <td>0.38</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>35.2</td>\n",
       "      <td>0.39</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>35.3</td>\n",
       "      <td>0.39</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>35.4</td>\n",
       "      <td>0.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>35.5</td>\n",
       "      <td>0.38</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>35.6</td>\n",
       "      <td>0.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>35.7</td>\n",
       "      <td>0.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>35.8</td>\n",
       "      <td>0.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>35.9</td>\n",
       "      <td>0.24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>36.0</td>\n",
       "      <td>0.28</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>36.5</td>\n",
       "      <td>0.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>37.0</td>\n",
       "      <td>0.43</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    position  current\n",
       "0       30.5     0.38\n",
       "1       30.6     0.42\n",
       "2       30.7     0.42\n",
       "3       30.8     0.42\n",
       "4       30.9     0.48\n",
       "5       31.0     0.43\n",
       "6       31.1     0.50\n",
       "7       31.2     0.56\n",
       "8       31.3     0.56\n",
       "9       31.4     0.53\n",
       "10      31.5     0.48\n",
       "11      32.0     0.46\n",
       "12      32.5     0.48\n",
       "13      33.0     0.40\n",
       "14      33.5     0.46\n",
       "15      33.6     0.48\n",
       "16      33.7     0.52\n",
       "17      33.8     0.53\n",
       "18      33.9     0.52\n",
       "19      34.0     0.46\n",
       "20      34.5     0.34\n",
       "21      35.0     0.34\n",
       "22      35.1     0.38\n",
       "23      35.2     0.39\n",
       "24      35.3     0.39\n",
       "25      35.4     0.36\n",
       "26      35.5     0.38\n",
       "27      35.6     0.36\n",
       "28      35.7     0.34\n",
       "29      35.8     0.30\n",
       "30      35.9     0.24\n",
       "31      36.0     0.28\n",
       "32      36.5     0.36\n",
       "33      37.0     0.43"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "standwaves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs position in a standing wave')"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8+PHVjN3MmtDQoUP7PBKcVVbLOotS94DFhWa3AW0RsS9wF3BtwbLxaXOuU4B/l7TbFjuL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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd38524d978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(standwaves['position'], standwaves['current'])\n",
    "plt.xlabel(r'position (cm)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs position in a standing wave')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "res = fit_sin(standwaves['position'], standwaves['current'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "est = res['fitfunc'](standwaves['position'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "std = np.sqrt(np.diag(res['rawres'][2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'current vs position in a standing wave')"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Xur/D3/8e10RRFNYMFQtr18L998Nll8Fpp/kdzYEqVXLNUWvXurvIo5BfwiJn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\n8zjGaoxJMT3a1uPeC1tRr3oGAtSrnsG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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd384fb3d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(standwaves['position'], standwaves['current'], label='experimental data')\n",
    "plt.plot(standwaves['position'], est,c=(1,0,0), label='fitted data')\n",
    "plt.xlabel(r'position (cm)')\n",
    "plt.ylabel('current (mA)')\n",
    "plt.legend()\n",
    "plt.title(r'current vs position in a standing wave')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fitted equation is $I(r) = .078 \\sin{(2.141\\theta - 8.921)} + .418$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = 0.078 \\pm .015$\n",
    "\n",
    "$\\omega = 2.141 \\pm .097$\n",
    "\n",
    "$\\phi = 8.921 \\pm 3.256$\n",
    "\n",
    "$c = .418 \\pm .011$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bonus\n",
    "Calculate the E-field and B-fields at the reflection plate? Is there any surface charge\n",
    "density 𝜎 on the plate? Submit a plot 𝜎 in the plane."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Submission\n",
    "This lab is an exploration lab, hence a full lab report is not required. Submit a homework style\n",
    "assignment answering all questions, rendering all plots, and calculating all quantiles assigned. All\n",
    "measurements must be reported with uncertainty – if you are using a plot to report your data use error\n",
    "bars for each measured point – and all calculated quantities must be reported with uncertainty as well."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}