Gauss's Law Exercise (part 5)

Electricity and Magnetism Level pending

A particle with charge q = + 10 q=+10 is position at N N ( x , y , z ) = ( 1 , 1 , 1 ) (x, y, z) =(1,1,1) at that orange dot as shown in figure. A closed conical frustum consists of a bottom surface, a top surface and curved surface. (Green)Bottom surface has equation x 2 + y 2 ≤ 16 x^{2}+y^{2}≤16 centered at orgin and parallel to X − Y X-Y plane. The(Blue) top surface has equation x 2 + y 2 ≤ 1 x^{2} + y^{2} ≤1 centered at ( 0 , 0 , 4 ) (0,0,4) and parallel to X − Y X-Y plane. And the third surface is curved surface. I have provided the view of figure from + Z +Z direction. Let the flux passing through these 3 surfaces be ϕ 1 , ϕ 2 , ϕ 3 \phi_{1}, \phi_{2}, \phi_{3} . Determine the following ratio : ϕ 1 ϕ 2 ϕ 3 ϕ 1 + ϕ 2 + ϕ 3 \frac{\phi_{1}\phi_{2}\phi_{3}}{\phi_{1}+\phi_{2}+\phi_{3}} Details and Assumptions 1) Electric permittivity ϵ o = 1 \epsilon_{o}=1


The answer is 0.445.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Steven Chase
Mar 14, 2020

Nice problem. Solution code is attached. The key here is that we only need to explicitly calculate the fluxes for the two disks. Then we can use Gauss's Law to get the flux for the curved surface. 

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import math

N = 5000

#####################################

q = 10.0

xq = 1.0
yq = 1.0
zq = 1.0

e0 = 1.0

#####################################

k = 1.0/(4.0*math.pi*e0)

phi1 = 0.0
phi2 = 0.0

#####################################
#####################################
#####################################
#####################################

# Bottom Disk 

dr = 4.0/N
dtheta = 2.0*math.pi/N

nx = 0.0
ny = 0.0
nz = -1.0

r = 0.0

while r <= 4.0:

    theta = 0.0

    while theta <= 2.0*math.pi:

        x = r*math.cos(theta)
        y = r*math.sin(theta)
        z = 0.0

        dS = r*dr*dtheta

        Dx = x - xq
        Dy = y - yq
        Dz = z - zq

        D = math.sqrt(Dx**2.0 + Dy**2.0 + Dz**2.0)

        ux = Dx/D
        uy = Dy/D
        uz = Dz/D

        E = k*q/(D**2.0)

        Ex = E * ux
        Ey = E * uy
        Ez = E * uz

        dot = Ex*nx + Ey*ny + Ez*nz

        dphi = dot * dS

        phi1 = phi1 + dphi

        theta = theta + dtheta

    r = r + dr

##################################################################
##################################################################

# Top Disk 

dr = 1.0/N
dtheta = 2.0*math.pi/N

nx = 0.0
ny = 0.0
nz = 1.0

r = 0.0

while r <= 1.0:

    theta = 0.0

    while theta <= 2.0*math.pi:

        x = r*math.cos(theta)
        y = r*math.sin(theta)
        z = 4.0

        dS = r*dr*dtheta

        Dx = x - xq
        Dy = y - yq
        Dz = z - zq

        D = math.sqrt(Dx**2.0 + Dy**2.0 + Dz**2.0)

        ux = Dx/D
        uy = Dy/D
        uz = Dz/D

        E = k*q/(D**2.0)

        Ex = E * ux
        Ey = E * uy
        Ez = E * uz

        dot = Ex*nx + Ey*ny + Ez*nz

        dphi = dot * dS

        phi2 = phi2 + dphi

        theta = theta + dtheta

    r = r + dr

##################################################################
##################################################################

# Curved Surface
# Use Gauss Law

phi3 = 10.0 - phi1 - phi2

##################################################################
##################################################################

print N
print ""
print phi1
print phi2
print phi3
print ""
num = phi1 * phi2 * phi3
denom = phi1 + phi2 + phi3
print (num/denom)


#>>> 
#1000

#3.68046254839
#0.197988258618
#6.12154919299

#0.446070170861
#>>> ================================ RESTART ================================
#>>> 
#2000

#3.67901825264
#0.198158344657
#6.1228234027

#0.446371072157
#>>> ================================ RESTART ================================
#>>> 
#5000

#3.67625732908
#0.197985417747
#6.12575725317

#0.445860388935
>>>

1 Helpful 1 Interesting 1 Brilliant 0 Confused

@Steven Chase Sir Ha Ha Ha Yes i agree with you i do not know how to calculate flux analytically through that curved surface. I just calculate through two disc and subtract it from 10 10 .

A Former Brilliant Member - 1 year, 2 months ago

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@Steven Chase in your code what does N = 5000 N=5000 mean??

A Former Brilliant Member - 1 year, 2 months ago

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For example, the disk has a radial dimension and an angular dimension. Each one of those is divided into 5000 5000 parts. So each surface ends up being divided into 25 25 million pieces. I always check the result for multiple different resolutions (N values), to be sure that the answer converges nicely as N N increases.

Steven Chase - 1 year, 2 months ago

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@Steven Chase – Today I started to learn code . Therefore I was asking . Simply you are doing integration . How much time does it take for you to run the code???

A Former Brilliant Member - 1 year, 2 months ago

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@A Former Brilliant Member – That's good. You'll find it very useful. This code will take anywhere from a few seconds to a minute or two to run

Steven Chase - 1 year, 2 months ago

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@Steven Chase – @Steven Chase is there any website or any YouTube channel where phython programming is well explained??

A Former Brilliant Member - 1 year, 2 months ago

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