Electromagnetismo¶
Una partícula con una carga positiva de 1C, se mueve cun una rapidez de 100m/s en la direccion X+ hacia un campo magnético uniforme de 10 T en la dirección Z+
¿Cúal es la trayectoria que sigue la patícula?
De la ecuación de Lorens
$$m\frac{dv}{dt}=q(\vec{v} \times \vec{B})(1)$$Se asume $\vec{B}=B_0 \widehat{z}$
El producto cruz seria
$$\vec{v} \times \vec{B}=(v_y \widehat{x} - v_x \widehat{y})B (2)$$y reemplanzando resulta. $$\frac{d\vec{v}}{dt}=\frac{qB}{m}(v_z \widehat{x} - v_x \widehat{y})$$(3)
se obtienen dos velocidades, que van a representar el movimiento en el plano siguiendo la fuerza que siente la particula en el campo magnético.
Para x: $$\frac{dv_x}{dt}=\frac{qB}{m}v_y$$(4) $$v_y= \frac{m}{qB}\frac{dv_x}{dt}$$(5)
para y
$$\frac{dv_y}{dt}=\frac{qB}{m}v_x $$(6)
reemplazando $v_y$ en 5 resulta $$\frac{d^2 v_x}{dt^2}+(\frac{qB}{m})^2 v_x=0$$(7)
$$v_x(t)=v_0 sin(\frac{qB}{m}t)$$(8) $$v_y (t)=v_0 cos(\frac{qB}{m}t)$$(9)
la ecuación total de la velocidad etaría dad por,
$$v_{x}^{2} + v_{y}^{2}=v_{0}^{2} sin^{2}(\frac{qB}{m}t)+v_{0}^{2}cos^{2}(\frac{qB}{m}t)$$(10)
Luego el movimiento perpendicular es compuesto de dos movimientos armónicos independientes dado por,
$$x(t)=\frac{mv_0}{qB} sin (\frac{qB}{m}t) + x_0$$(11)
$$y(t)=-\frac{mv_0}{qB} cos(\frac{qB}{m}t)+y_{0}$$(12)
In [3]:
from __future__ import division
import numpy as np
import string
import random
import sys
import pylab as p
import mpl_toolkits.mplot3d.axes3d as p3
from matplotlib import pyplot as p
from matplotlib import animation
from pip._vendor.distlib.compat import raw_input
%matplotlib auto
masa = 1
carga = 1
posicion = np.array([-1000, 0, -1000])
velocidad = np.array([0, 10, 0])
B = np.array([0, 0, 0.09])
E = np.array([0, 0, 0.09])
dt = 0
paso = 0.05
while True:
texto = raw_input("ingrese la masa:\n")
masa = float(texto)
if masa == 0:
print("entrada mala.")
else:
break
while True:
texto = raw_input("ingrese la carga:\n")
carga = float(texto)
if masa == 0:
print("entrada mala.")
else:
break
while True:
texto = raw_input("ingrese la posicion de la carda separado por espcios:\n")
ind = texto.split(" ")
if len(ind) != 3:
print("entrada mala.")
else:
posicion[0] = float(ind[0])
posicion[1] = float(ind[1])
posicion[2] = float(ind[2])
break
while True:
texto = raw_input("ingrese la velocidad inicial de la carda separado por espcios:\n")
ind = texto.split(" ")
if len(ind) != 3:
print("entrada mala.")
else:
velocidad[0] = float(ind[0])
velocidad[1] = float(ind[1])
velocidad[2] = float(ind[2])
break
while True:
texto = raw_input("ingrese el campo electrico separado por espcios:\n")
ind = texto.split(" ")
if len(ind) != 3:
print("entrada mala.")
else:
E[0] = float(ind[0])
E[1] = float(ind[1])
E[2] = float(ind[2])
break
while True:
texto = raw_input("ingrese el campo magnetico separado por espcios:\n")
ind = texto.split(" ")
if len(ind) != 3:
print("entrada mala.")
else:
B[0] = float(ind[0])
B[1] = float(ind[1])
B[2] = float(ind[2])
break
color = "m"
if carga < 0:
color = "b"
tamanoCaja = 1000
titulo = "Masa = " + str(masa) + "; carga = " + str(carga) + "; Vo= " + str(velocidad)
fig = p.figure()
fig.suptitle(titulo, fontsize=8, fontweight='bold')
ax = p3.Axes3D(fig)
ax.set_xlim3d({- tamanoCaja, tamanoCaja})
ax.set_ylim3d({- tamanoCaja, tamanoCaja})
ax.set_zlim3d({- tamanoCaja, tamanoCaja})
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
fig.suptitle(ax)
contador = 0
def anima(a):
global dt, posicion, velocidad, paso, carga, B, E, masa, color, contador
contador += 1
# ax.elev = 90
dt += paso
t = 0.5 * carga * B * dt / masa
t_2 = np.linalg.norm(t) * np.linalg.norm(t)
s = 2 * t / (1 + t_2)
v_minus = velocidad + 0.5 * carga * E * dt / masa
v_prime = v_minus + np.cross(v_minus, t)
v_plus = v_minus + np.cross(v_prime, s)
velocidad = v_plus + 0.5 * carga * E * dt / masa
posicion = posicion + velocidad * dt
ax.scatter(posicion[0], posicion[1],
posicion[2], s=1, c=color, marker="o")
if posicion[0] < - tamanoCaja or posicion[0] > tamanoCaja:
sys.exit()
if posicion[1] < - tamanoCaja or posicion[1] > tamanoCaja:
sys.exit()
if posicion[2] < - tamanoCaja or posicion[2] > tamanoCaja:
sys.exit()
print(contador, posicion)
# fig.savefig("imgs/" + str(i) + ".png"
anim = animation.FuncAnimation(fig, anima, frames=10000, interval=1)
p.show()
Using matplotlib backend: TkAgg ingrese la masa: 1 ingrese la carga: 1 ingrese la posicion de la carda separado por espcios: 0 0 0 ingrese la velocidad inicial de la carda separado por espcios: 100 0 0 ingrese el campo electrico separado por espcios: 0 0 0 ingrese el campo magnetico separado por espcios: 0 0 10 1 [ 4.41176471 -2.35294118 0. ] 2 [ 5.94117647 -12.23529412 0. ] 3 [ -7.64705882 -18.58823529 0. ] 4 [-16.11764706 -0.47058824 0. ] 5 [8.30129125 4.88809182 0. ] 6 [ 2.96678071 -24.63381525 0. ] 7 [-23.54684913 -1.7858956 0. ] 8 [15.52345216 6.7879925 0. ] 9 [ -6.77068028 -32.30128556 0. ] 10 [-18.78634235 16.23348409 0. ] 11 [ 25.63683043 -16.19461669 0. ] 12 [-34.35814996 -16.97071455 0. ] 13 [18.92145891 20.26211074 0. ] 14 [ -8.61248314 -44.09534443 0. ] 15 [-17.36307916 30.39242013 0. ] 16 [ 28.26271264 -35.32125518 0. ] 17 [-46.26173758 5.55791213 0. ] 18 [43.55181842 -0.23217932 0. ] 19 [-45.66858295 -32.86222651 0. ] 20 [27.8127596 34.96470249 0. ] 21 [-17.75918534 -59.63021194 0. ] 22 [-7.95563901 49.93205565 0. ] 23 [ 21.06817027 -61.34515897 0. ] 24 [-45.23951199 38.67129604 0. ] 25 [ 52.88950111 -38.76058879 0. ] 26 [-68.65070109 7.36943669 0. ] 27 [66.03183588 -1.88335945 0. ] 28 [-70.73884988 -31.77950499 0. ] 29 [57.24487197 36.37595658 0. ] 30 [-52.05376453 -66.35577201 0. ] 31 [30.24244929 64.99236536 0. ] 32 [-18.71979425 -87.33195544 0. ] 33 [-7.21938139 77.26677027 0. ] 34 [ 20.61320246 -90.43936342 0. ] 35 [-46.25070937 71.28329206 0. ] 36 [ 57.3605613 -75.90605739 0. ] 37 [-78.99927778 49.11791747 0. ] 38 [ 84.71244501 -47.31066659 0. ] 39 [-99.89876924 15.48821535 0. ] 40 [ 98.45094236 -10.15143966 0. ] 41 [-106.10334672 -23.66228045 0. ] 42 [97.06043379 29.48354053 0. ] 43 [-97.33549411 -62.35993074 0. ] 44 [81.37324 65.9490259 0. ] 45 [-75.38545682 -95.45639833 0. ] 46 [53.97267421 94.71845262 0. ] 47 [ -43.46066197 -119.13136618 0. ] 48 [ 18.52428715 112.72606002 0. ] 49 [ -5.52716813 -131.09052576 0. ] 50 [-20.85065674 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0. ] 79 [-114.75563732 157.84356348 0. ] 80 [ 125.68284838 -161.82710235 0. ] 81 [-148.465956 136.27930996 0. ] 82 [ 157.12245092 -137.06246431 0. ] 83 [-177.36487513 108.58612353 0. ] 84 [ 183.24407075 -106.73036015 0. ] 85 [-200.50625557 75.91872156 0. ] 86 [203.23590813 -72.0416287 0. ] 87 [-217.21251679 39.52952339 0. ] 88 [216.55317473 -34.27647133 0. ] 89 [-227.06907451 0.71352692 0. ] 90 [222.90788257 5.26743711 0. ] 91 [-229.91004577 -39.24013272 0. ] 92 [222.25100859 45.31979232 0. ] 93 [-225.79739104 -79.08933752 0. ] 94 [214.74910424 84.67508319 0. ] 95 [-214.99545135 -117.67267919 0. ] 96 [200.75718898 122.22259392 0. ] 97 [-197.94259485 -153.93584316 0. ] 98 [180.78951463 156.96926976 0. ] 99 [-175.22143196 -186.95107844 0. ] 100 [155.48953274 188.05593193 0. ] 101 [-147.52880593 -215.93031341 0. ] 102 [125.60015023 214.76746149 0. ] 103 [-115.64652733 -240.23258309 0. ] 104 [ 91.93512966 236.53765055 0. ] 105 [ -80.41360189 -259.3664881 0. ] 106 [ 55.35228286 252.949454 0. ] 107 [ -42.70050601 -272.98842312 0. ] 108 [ 16.72892393 263.73137608 0. ] 109 [ -3.38589402 -280.89729935 0. ] 110 [-23.06010991 268.75070186 0. ] 111 [ 36.66402394 -283.02645229 0. ] 112 [-63.16169423 268.00424195 0. ] 113 [ 76.61331544 -279.43337703 0. ] 114 [-102.76084407 261.60720454 0. ] 115 [ 115.67135147 -270.2878753 0. ] 116 [-141.09435369 249.78074773 0. ] 117 [ 153.10508332 -255.85913525 0. ] 118 [-177.46174073 232.83870067 0. ] 119 [ 188.24869014 -236.50219969 0. ] 120 [-211.23363823 211.17387664 0. ] 121 [ 220.51076081 -212.64421393 0. ] 122 [-241.85781544 185.24433792 0. ] 123 [ 249.37920551 -184.77078238 0. ] 124 [-268.86303388 155.55991167 0. ] 125 [ 274.42411161 -153.4127045 0. ] 126 [-291.86095967 122.66920009 0. ] 127 [ 295.29876948 -119.13330706 0. ] 128 [-310.54635987 87.14727673 0. ] 129 [311.73909636 -82.51654077 0. ] 130 [-324.69581032 49.58421455 0. ] 131 [323.5616837 -44.15596596 0. ] 132 [-334.1651369 10.57455071 0. ] 133 [330.66068539 -4.64471374 0. ] 134 [-338.88580241 -29.29224262 0. ] 135 [333.00375335 35.43352406 0. ] 136 [-338.86043902 -69.44024297 0. ] 137 [330.62721323 75.51245097 0. ] 138 [-334.1577117 -109.31450483 0. ] 139 [323.63065779 115.05006671 0. ] 140 [-324.90668183 -148.38743843 0. ] 141 [312.17111926 153.53453056 0. ] 142 [-311.29082449 -186.16416637 0. ] 143 [296.45696545 190.48902136 0. ] 144 [-293.54183554 -222.18689851 0. ] 145 [276.74164794 225.47564011 0. ] 146 [-271.93334884 -256.03837622 0. ] 147 [253.31741438 258.09841078 0. ] 148 [-246.77466781 -287.34444538 0. ] 149 [226.50908156 288.00544172 0. ] 150 [-218.40460054 -315.77582288 0. ] 151 [196.66795131 314.89031439 0. ] 152 [-187.18547345 -341.04912512 0. ] 153 [164.16593744 338.49276915 0. ] 154 [-153.4973855 -362.92722861 0. ] 155 [129.38995957 358.59875845 0. ] 156 [-117.7327527 -381.21903336 0. ] 157 [ 92.73664807 375.03993772 0. ] 158 [ -80.29118179 -395.7786986 0. ] 159 [ 54.6073937 387.69266293 0. ] 160 [ -41.57470211 -406.50441869 0. ] 161 [ 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