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File:Magnetic field of an idealized quadrupole with forces.svg

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Summary

Description
English: Magnetic field of an idealized quadrupole with forces
Русский: Магнитное поле и силы в квадрупольном магните
Date
Source python/matplotlib
Author Andre.holzner
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SVG development
InfoField
 
The source code of this SVG is invalid due to an error.
 
This W3C-invalid vector image was created with an unknown SVG tool.

Other information

xpoints = arange(-5,5,0.05)
ypoints = arange(-5,5,0.05)
X,Y = meshgrid(xpoints, ypoints)

circularMask = False

areaRadius = 4

# order of the magnet
n = 2

def func(x,y):
    # the function to draw
    return ((x + 1j * y)**(n)).real

func = vectorize(func)

V = func(X,Y)

# mask points which we don't want to draw
if circularMask:
    # circular mask
    distance = sqrt(X**2 + Y**2)
    V = ma.masked_where(distance > areaRadius, V)
else:
    # polygonal mask

    # principal directions are at  (i + 0.5) / (2n) * 2pi
    # 
    for i in range(2*n):
        angle = (i + 0.5) / float(2*n) * 2*pi
    
        # define a straight angle perpendicular to angle
        # mask all points on one side of this line
        anchor_x = areaRadius * cos(angle)
        anchor_y = areaRadius * sin(angle)

        normal_x = cos(angle)
        normal_y = sin(angle)
        
        def acceptFunc(x,y):
            value = (x - anchor_x) * normal_x + (y - anchor_y) * normal_y
            return value > 0
        
        acceptFunc = vectorize(acceptFunc)
        
        V = ma.masked_where(acceptFunc(X,Y), V)        

if True:
    # levels equidistant in function value
    vmax = V.max()
    V /= vmax

    levels = arange(-2,2,0.05)

else:
    # levels equidistant on x and y axis

    # determine the levels to draw from values on one of the axes

    levels = [ float(func(x,0)) for x in arange(min(xpoints), max(xpoints),0.50) ] + \
        [ float(func(0,y)) for y in arange(min(ypoints), max(ypoints),0.50) ]
    levels = sorted(list(set(levels)))

    vmax = 1
    
figure(figsize=(6,6)); 
Q = contour(X,Y, V, colors=  'black', linestyles = 'solid', 
    levels = levels
)
# axis([-5,5,-5,5])
xlabel("x coordinate")
ylabel("y coordinate")

# mask points which we don't want to draw
if not circularMask:
    # polygonal mask

    # principal directions are at  (i + 0.5) / (2n) * 2pi
    # 
    for i in range(2*n):
        angle = (i + 0.5) / float(2*n) * 2*pi
        
        if i % 2:
            label = "N"
            color = 'red'
        else:
            label = "S"
            color = 'green'
        
        anchor_x = 1.1 * areaRadius * cos(angle)
        anchor_y = 1.1 * areaRadius * sin(angle)

        text(anchor_x, anchor_y, label, size = 20, color = color,
             horizontalalignment='center',
             verticalalignment='center')

#----------------------------------------
if n == 2:
    # quadrupole, draw some examples of force on charged particle

    # find kth level line on axes (x = 0 and y = 0)

    # the potential function is >= 0 on the x axis and <= 0 on the y axis
    # for a quadrupole
    lev = sorted(list(levels[levels >= 0]))[4]

    # find distance of this level on axis from origin
    # (exploit the 90 degree symmetry of the field)
    dist = fsolve(lambda x: func(x,0) / vmax - lev,3)[0]
    
    # rotation by +90 degrees
    rotMatrix = array([[0,-1],[1,0]])
    invRotMatrix = rotMatrix.T
    
    arrowLength = 1.5
    
    arrowStart = array([dist, 0])
    origArrowDir = array([0, arrowLength])
    
    bfieldLabelPosOffset = array([0.3, 0.5 * arrowLength])
    forceLabelPosOffset = array([0.5 * arrowLength, -0.3])
    
    for i in range(4):
        
        arrowDir = origArrowDir[:]
        for j in range(i):
            arrowDir = rotMatrix.dot(arrowDir)
        
        # take into account quadrupole structure
        arrowDir *= (-1)**i
        
        # draw arrow for the B field
        arrow(arrowStart[0],arrowStart[1], arrowDir[0], arrowDir[1],head_width=0.3, head_length=0.3, color = 'red')
    
        # add a label for the B field
        textPos = arrowStart + (-1)**i * bfieldLabelPosOffset
        text(textPos[0], 
             textPos[1], "B", size = 20, color = 'red',
             horizontalalignment='center',
             verticalalignment='center')

        # draw the arrow for the force
        arrowDir2 = invRotMatrix.dot(arrowDir)
    
        arrow(arrowStart[0], arrowStart[1], arrowDir2[0], arrowDir2[1],head_width=0.3, head_length=0.3, color = 'blue')
        
        # label for the force
        textPos = arrowStart + (-1)**i * forceLabelPosOffset
        text(textPos[0], 
             textPos[1], "F", size = 20, color = 'blue',
             horizontalalignment='center',
             verticalalignment='center')
        #----------
        # prepare next iteration
        arrowStart = rotMatrix.dot(arrowStart)
        
        bfieldLabelPosOffset = rotMatrix.dot(bfieldLabelPosOffset)
        forceLabelPosOffset = rotMatrix.dot(forceLabelPosOffset)

Licensing

Andre.holzner, the copyright holder of this work, hereby publishes it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Attribution: Andre.holzner
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
You may select the license of your choice.

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16 December 2012

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current21:12, 16 December 2012Thumbnail for version as of 21:12, 16 December 2012540 × 540 (171 KB)Andre.holznerUploading a self-made file using File Upload Wizard

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