ElectroDynamics, by James Rohlf and Kevin Reiss, Version 1.0, Copyright 2021, All Rights Reserved.

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## Excerpts

Excerpts

The electric dipole plays a fundamental role in electromagnetism, from statics to radiation, and the electric potential is an invaluable concept for its visualization and interpretation.

Out[193]=

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Calculate the ratio of electric to gravitational force for an electron-proton interaction.

In[28]:=

ClearAll["Global`*"];UnitConvert[()/([][])]

2

4π

Out[28]=

2.269×

39

10

Draw the line charge integration geometry off the symmetry axis.

Out[495]=

Calculate the electric field at (x, 0, z) from the line of charge above.

In[8]:=

r={x,0,z};r′={x′,0,0};ℛ=r-r′;$Assumptions={L>0,z∈,z≠0,x∈};Ε=x′//Together//FullSimplify

λ

4π

ε

0

L/2

∫

-L/2

ℛ

3/2

(ℛ.ℛ)

Out[9]=

,0,

-+4++4λ

2

(L-2x)

2

z

2

(L+2x)

2

z

2π+8(-+)+16

4

L

2

L

2

x

2

z

2

(+)

2

x

2

z

ε

0

2x+4-+4+L+4++4λ

2

(L-2x)

2

z

2

(L+2x)

2

z

2

(L-2x)

2

z

2

(L+2x)

2

z

4πz+8(-+)+16

4

L

2

L

2

x

2

z

2

(+)

2

x

2

z

ε

0

Calculate the electric field for .

L=0.1,x=0.4,z=.3,andλ=1

In[11]:=

UnitConvert[Ε/.{L->0.1,x->0.4,z->.3,λ->1/,,0->0/},/]

ε

0

Out[11]=

{,,}

Plot the electric field in the x-z plane.

In[26]:=

VectorPlot[Ε[[{1,3}]]/.{L2,λ1,1},{x,-2,2},{z,-2,2},EpilogLine[{{-1,0},{1,0}}],]

ε

0

Out[26]=

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Draw the geometry for dividing a solid ball into disks.

Out[148]=

Calculate the electric field outside a uniform ball of charge, by adding together disks.

In[194]:=

ClearAll["Global`*"];$Assumptions={z′∈Reals,z′>0,r∈Reals,r>R,R>0};ρ=;(1--+)z′

Q

(4/3)π

3

R

ρ

2

ε

0

R

∫

-R

r-z′

2

R

2

z′

2

(r-z′)

Out[194]=

Q

4π

2

r

ε

0

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Calculate the electric field of an ideal dipole.

Transform the electric field to Cartesian coordinates.

Verify that the coordinate transformation produces the same result as taking the gradient in Cartesian.

Verify that the curl is zero.

Draw the electric field lines for the ideal dipole.

Draw the electric field lines for an ideal dipole in 3D.

■

Wolfram Research (2021), StreamPlot3D, Wolfram Language function, https://reference.wolfram.com/language/ref/StreamPlot3D.html.

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Draw the boundary conditions for the solution of Laplace’s Equation.

Calculate the full form of the potential inside the above region.

Plot the solution for a = 2 and b = 1, keeping 500 terms.

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The following demonstration solves Laplace’s equation for arbitrary boundary conditions on various geometries. A certain sinusoidal boundary condition produces a solution in the shape of a “Pringle” chip.

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Calculate the magnetic field due to a circular loop of radius R and current ℐ at a distance z from the center along the axis of symmetry.

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### 2.9 Summary of B, A, and J

2.9 Summary of B, A, and J

Here is a summary of how to calculate one quantity from another in magneto-statics. The hardest conversion to do is to get A from B because there is no easy recipe for undoing a curl. See, for example, Section 2.5.

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A dielectric ball placed in a uniform external electric field is an important example that will appear again in the discussion of magnetic fields in matter. The result is a uniform electric field inside the ball, which contributes to the external field as a pure dipole. It is described in Section 3.4.

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When squaring the electric field, one may note that the cross term is twice as large as the second term squared, and has the opposite sign. The Poynting vector is

Make a polar plot of the Poynting vector (energy flux).

Plot the 3D radiation pattern.

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Make a polar plot of the radiation pattern for β=.9.

Calculate 1-β.

Calculate the radiated power per proton.

Calculate the radiated power per beam.

Calculate the radiated energy per proton per revolution.

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Make a polar plot of the electric quadrupole energy flux.

Make a 3D plot of the electric quadrupole energy flux.

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### 8.5 Magnetic Quadrupole Radiation

8.5 Magnetic Quadrupole Radiation

A magnetic quadruple can be constructed out of 2 out-of-phase magnetic dipoles. Consider 2 current loops of radius d separated by distance b.

This time the first 2 terms of the expansion of the exponential both give zero and we must use the 3rd term. We have separated the current into 2 loops with opposite sign.

Calculate the magnetic quadrupole angular term.

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Display the contravariant form of the field tensor.

Display the covariant form of the field tensor.

Transform the field tensor.

Transform back.