vdP_jl.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# vdP.ipynb\n",
    "\n",
    "### by Hezy Amiel\n",
    "\n",
    "find the sheet resistance Rs from R1 = R_AB,CD and R2 = R_BC,AD\n",
    "the solution is from the equation in a paper by L. J. van der Pauw:\n",
    "\"A method of measuring the resistivity and hall coefficients of Lamellae of arbitrary shape\", Philips Technical Review, Vol 26, 220.\n",
    "\n",
    "$e^{-\\pi R_{AB,CD}/R_s}+e^{-\\pi R_{BC,DA}/R_s}=1$\n",
    "\n",
    "so we dfine the function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fun (generic function with 2 methods)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fun(Rs) = @. exp(-π * R1 / Rs) + exp(-π * R2 / Rs) - 1\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The resistance $R_{AB,CD}$ and $R_{BC,AD}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "R1 = 100\n",
    "R2 = 50\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### plot the fuction\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using Plots\n",
    "R = LinRange(0, 10 * sqrt(R1 * R2), 200)\n",
    "p = plot(R, fun(R))\n",
    "p = hline!([0], linestyle=:dash)\n",
    "display(p)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### find the solution\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "326.4251302636497"
     ]
    },
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using Roots\n",
    "root = find_zero(fun, Rs)\n",
    "print(root)\n",
    "p = vline!([root], linestyle=:dash)"
   ]
  }
 ],
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