Wolfram Cloud Document

In[]:=

Dist[x_]:=Sqrt[Total[x^2]]qposr[tr_]:={0,0,v0Sin[omegatr]};(*pointchargepositionretardedtime*)rPrime[tr_]:=D[qposr[tr],tr];(*pointchargevelocityretardedtime*)(*Observationpoint*)r={x,y,z};(*VectorpotentialAatobservationpoint,usingtrforretardedtime*)A[tr_]:=(μ0q/(4Pi))qvr[tr]/Dist[r-rPrime[tr]];(*CalculatethepartialderivativeofAwithrespecttotforEfieldfromthevectorpotenmtial*)EA=-D[A[tr]/.tr->t-Dist[r-rPrime[t]]/c,t];(*DefinethescalarpotentialPhiatobservationpoint,usingtrforretardedtime*)Phi[tr_]:=(1/(4Piϵ0))q/Dist[r-rPrime[tr]];(*Scalarpotentialϕatobservationpoint*)ϕ:=Phi[tr]/.tr->t-Dist[r-rPrime[t]]/c;Eϕ:=-Grad[ϕ,{x,y,z}]Efield:=Eϕ+EA

EfieldNull=Efield/.t->0,x->0,y->0,μ0

1

2

c

ϵ0

(*lookonlyat"antenna"null(alongz)*)EfieldNullNaturalUnits=Assuming[z>2&&zϵReals,EfieldNull/.{q->1,v0->1,c->1,omega->1,ϵ0->1}]//FullSimplify

Out[]=

0,0,-

3

omega

q

2

v0

Cos

omega

2

(-omegav0+z)

c

z-omegav0Cos

omega

2

(-omegav0+z)

c

Sin

omega

2

(-omegav0+z)

c



4

2

c

πϵ0

3/2

2

z-omegav0Cos

omega

2

(-omegav0+z)

c



-

2

omega

qv0Sin

omega

2

(-omegav0+z)

c



4

2

c

πϵ0

2

z-omegav0Cos

omega

2

(-omegav0+z)

c



+

qz-omegav0Cos

omega

2

(-omegav0+z)

c

1+

2

omega

v0(-omegav0+z)Sin

omega

2

(-omegav0+z)

c



c

2

(-omegav0+z)

4πϵ0

3/2

2

z-omegav0Cos

omega

2

(-omegav0+z)

c





Out[]=

0,0,

2

(z-Cos[1-z])

-1+z+1+-1+

2

(-1+z)

zSin[1-z]

4π(-1+z)

3

(z-Cos[1-z])



Plot[EfieldNullNaturalUnits*z,{z,100,1000}](*TimesztoshowsupposednullEfieldof"antenna"fallsoffas1/r*)

Out[]=

In[]:=

EfieldDenominator:=4π(-1+z)

3

(z-Cos[1-z])

//Expand

In[]:=

Plot[EfieldDenominator^(1/3),{z,1,100000}](*Cuberoottoshowitprovidesinversecubelaw(ignoringnumerator)*)

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In[]:=

EfieldNumerator:=

2

(z-Cos[1-z])

-1+z+1+-1+

2

(-1+z)

zSin[1-z]

In[]:=

Plot[EfieldNumerator^(1/2),{z,1,100000}](*Squreroottoshownumeratorcounteractsdenominatortoprovidesimpleinverselaw*)

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