File:Cylindrical-magnet-force-diagram.svg
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Original file (SVG file, nominally 720 × 540 pixels, file size: 57 KB)
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Summary
DescriptionCylindrical-magnet-force-diagram.svg |
English: Exactly computed force between two axially aligned identical cylindrical bar-magnets vs. distance between the magnets. Various graphs are shown for different lengths L of the magnets. The force is given in units of where M is the magnetization and R the radius. The force decreases sharply at small distances z. |
Date | |
Source | Own work |
Author | Geek3 |
Other versions | Cylindrical-magnet-force-diagram logscale.svg log-scale version |
SVG development InfoField | This plot was created with Matplotlib. |
Source code InfoField | Python code#!/usr/bin/python
# -*- coding: utf8 -*-
import numpy as np
import scipy.special as sp
import matplotlib.pyplot as plt
import matplotlib as mpl
from math import *
mpl.style.use("classic")
# fix elliptic integrals for negative argument in case of old scipy version
if sp.ellipe(-1) > 0:
E = sp.ellipe
K = sp.ellipk
else:
def E(m):
if m >= 0.:
return sp.ellipe(m)
else:
return sp.ellipe(-m / (1. - m)) * sqrt(1. - m)
def K(m):
if m >= 0.:
return sp.ellipk(m)
else:
return sp.ellipk(-m / (1. - m)) / sqrt(1. - m)
def force_between_disks(z):
'''
Exact formula for the force between two homogeneously charged round disks
aligned on their axis of symmetry.
z is the distance relative to the disk radius.
The force is returned in units of Q^2 / (4pi epsilon_0 R^2)
in case of an electric charge Q on each disk.
The solution requires elliptical integrals
'''
if z == 0.:
return 2.
elif z > 0.:
s = 1.0
else:
s = -1.0
z = -z
m = 4 / (4. + z**2)
return s * (2 + 4/pi * z / sqrt(m) * (E(m) - K(m)))
def force_between_magnets(z, R, L):
'''
Exact formula for the force between two axially aligned identical
cylindrical magnets, as long as they are homogeneously magnetized.
'''
zR = z / R
F = force_between_disks(zR)
F += force_between_disks(zR + 2*L / R)
F -= 2 * force_between_disks(zR + L / R)
return F
mpl.rcParams['font.sans-serif'] = 'DejaVu Sans'
mpl.rc('mathtext', default='regular')
mpl.rc('lines', linewidth=2.4)
colors = ['#0000ff', '#00aa00', '#ff0000', '#ee9900', '#cccc00']
L = [(r'$\infty$', float('inf')), ('4R', 4.), ('2R', 2.), ('R', 1.), ('R/2', 0.5)]
plt.figure()
zmax = 4
zspace = np.linspace(0., zmax**0.5, 5001)**2
for i in range(len(L)):
if L[i][1] == float('inf'):
f = lambda z: force_between_disks(z)
else:
f = lambda z: force_between_magnets(z, 1., L[i][1])
plt.plot(zspace, [f(z) for z in zspace], '-',
color=colors[i], label=r'L = ' + L[i][0], zorder=-i-len(L))
plt.plot(0, f(0), 'o', color=colors[i], mew=1.2, zorder=-i)
plt.xlabel('z / R')
plt.ylabel(r'$F\ [\pi/4\;\mu_0M^2R^4]$')
plt.title('Force between two cylindrical magnets with magnetization M,\nlength L, radius R and axial end-to-end distance z')
plt.legend(loc='upper right')
plt.xlim(-0.05, zmax)
plt.ylim(0, 2.1)
plt.grid(True)
plt.tight_layout()
plt.savefig('Cylindrical-magnet-force-diagram.svg')
plt.figure()
zmax = 20
zspace = np.linspace(0., zmax**0.5, 5001)**2
for i in range(len(L)):
if L[i][1] == float('inf'):
f = lambda z: force_between_disks(z)
else:
f = lambda z: force_between_magnets(z, 1., L[i][1])
plt.plot(zspace, [f(z) for z in zspace], '-',
color=colors[i], label=r'L = ' + L[i][0], zorder=-i-len(L))
plt.plot(0, f(0), 'o', color=colors[i], mew=1.2, zorder=-i)
plt.xlabel('z / R')
plt.ylabel(r'$F\ [\pi/4\;\mu_0M^2R^2]$')
plt.title('Force between two cylindrical magnets with\nmagnetization M, length L, radius R and axial distance z')
plt.gca().set_yscale('log')
plt.legend(loc='upper right')
plt.xlim(-0.5, zmax)
plt.ylim(1e-5, 2.5)
plt.grid(True)
plt.tight_layout()
plt.savefig('Cylindrical-magnet-force-diagram_logscale.svg')
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
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- Under the following conditions:
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Items portrayed in this file
depicts
24 September 2017
image/svg+xml
bb0695fd36778163d9ce4493230423b8e6f82daa
57,918 byte
540 pixel
720 pixel
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 15:01, 23 March 2021 | 720 × 540 (57 KB) | Geek3 | unit must contain R^2 | |
21:14, 1 April 2019 | 720 × 540 (58 KB) | Geek3 | title more specific | ||
17:54, 24 September 2017 | 720 × 540 (56 KB) | Geek3 | User created page with UploadWizard |
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Width | 576pt |
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Height | 432pt |