Supplement2 to eucidean 4dimensional electromagnetism
20/05/2014 23:32
Supplement2 to eucidean 4dimensional electromagnetism
Here I am writing a small supplement to euclidean 4dimensional electromagnetism that is most about the magnetical vector potential.
Axy=∫Bxydy=µ0∬jx(dy)2 Axz=∫Bxzdz=µ0∬jx(dz)2
Axct=∫Bxctcdt=µ0∬jx(cdt)2
Ax=Axy+Axz-Axct=µ0∬jx((dy)2+(dz)2-(cdt)2)
Ayx=∫Byxdx=µ0∬jy(dx)2 Ayz=∫Byzdz=µ0∬jy(dz)2
Ayct=∫Byctcdt=µ0∬jy(cdt)2
Ay=Ayx+Ayz-Ayct=µ0∬jy((dx)2+(dz)2-(cdt)2)
Azx=∫Bzxdx=µ0∬jz(dx)2 Azy=∫Bzydy=µ0∬jz(dy)2
Azct=∫Bzctcdt=µ0∬jz(cdt)2
Az=Azx+Azy-Azct=µ0∬jz((dx)2+(dy)2-(cdt)2)
Usx/c=∫(Esx/c)dx=µ0∬(ρ0vt)(dx)2
Usy/c=∫(Esy/c)dy=µ0∬(ρ0vt)(dy)2
Usz/c=∫(Esz/c)dz=µ0∬(ρ0vt)(dz)2
Us/c=Usx/c+Usy/c+Usz/c=µ0∬(ρ0vt)((dx)2+(dy)2+(dz)2)
A42=Ax2+Ay2+Az2+(Us/c)2 A4=(-Ax;-Ay;-Az;(Us/c))
Where Ax is the magnetical vector potential from currents flowing in x-direction
Axy is the magnetical vector potential from currents flowing in x-direction in the y-direction , Axz is the magnetical vector potential from currents flowing in x-direction in the z-direction , Axct is the magnetical vector potential from currents flowing in x-direction in the time dimension , Ay is the magnetical vector potential from currents flowing in y-direction , Ayx is the magnetical vector potential from currents flowing in y-direction in the x-direction , Ayz is the magnetical vector potential from currents flowing in y-direction in the z-direction , Ayct is the magnetical vector potential from currents flowing in y-direction in the time dimension , Az is the magnetical vector potential from currents flowing in z-direction , Azx is the magnetical vector potential from currents flowing in z-direction in the x-direction , Azy is the magnetical vector potential from currents flowing in z-direction in the y-direction , Azct is the magnetical vector potential from currents flowing in z-direction in the time dimension , Us/c is the electrostatic potential/c , Usx/c is the electrostatic potential/c in x-direction , Usy/c is the electrostatic potential/c in y-direction , Usz/c is the electrostatic potential/c in z-direction , A4 is the 4dimensional electromagnetical vector potential.
Φxy=∬Bxydydx=∫Axydx Φxz=∬Bxzdzdx=∫Axzdx
Φyx=∬Byxdxdy=∫Ayxdy Φyz=∬Byzdzdy=∫Ayzdy
Φzx=∬Bzxdxdz=∫Azxdz Φzy=∬Bzydydz=∫Azydz
Where Φxy is the magnetic flux from currents flowing in x-direction in the xy-plane , Φxz is the magnetic flux from currents flowing in x-direction in the xz-plane , Φyx is the magnetic flux from currents flowing in y-direction in the xy-plane , Φyz is the magnetic flux from currents flowing in y-direction in the yz-plane , Φzx is the magnetic flux from currents flowing in z-direction in the xz-plane , Φzy is the magnetic flux from currents flowing in z-direction in the yz-plane. Bxy is the magnetic flux density from currents flowing in x-direction in the y-direction , Bxz is the magnetic flux density from currents flowing in x-direction in the z-direction , Byx is the magnetic flux density from currents flowing in y-direction in the x-direction , Byz is the magnetic flux density from currents flowing in y-direction in the z-direction , Bzx is the magnetic flux density from currents flowing in z-direction in the x-direction , Bzy is the magnetic flux density from currents flowing in z-direction in the y-direction , Esx/c is the electrostatic field/c in the x-direction , Esy/c is the electrostatic field/c in the y-direction , Esz/c is the electrostatic field/c in the z-direction.
Ux=∫Exdx=∫(d(Usxcdt)/(cdT))-dΦyx/dT-dΦzx/dT=∫(d(Usxcdt)/(cdT))-∫(d(Ayxdy)/dT)-∫(d(Azxdz)/dT)=vtUsx/c+∫(dUsx/(cdT))cdt-vyAyx-∫(dAyx/dT)dy-vzAzx-∫(dAzx/dT)dz=vtµ0∬(ρ0vt)(dx)2+µ0∫(d(∬(ρ0vt)(dx)2)/dT)cdt-vyµ0∬jy(dx)2-µ0∫(d(∬jy(dx)2)/dT)dy-vzµ0∬jz(dx)2-µ0∫(d(∬jz(dx)2)/dT)dz
Ux=∫Exdx=∫(d(Usxcdt)/(cdT))-dΦyx/dT-dΦzx/dT=∫(d(Usxcdt)/(cdT))-∫(d(Ayxdy)/dT)-∫(d(Azxdz)/dT)=vtUsx/c+∫(dUsx/(cdT))cdt-vyAyx-∫(dAyx/dT)dy-vzAzx-∫(dAzx/dT)dz=vtµ0∬(ρ0vt)(dx)2+µ0∫(d(∬(ρ0vt)(dx)2)/dT)cdt-vyµ0∬jy(dx)2-µ0∫(d(∬jy(dx)2)/dT)dy-vzµ0∬jz(dx)2-µ0∫(d(∬jz(dx)2)/dT)dz
Uy=∫Eydy=∫(d(Usycdt)/(cdT))-dΦxy/dT-dΦzy/dT=∫(d(Usycdt)/(cdT))-∫(d(Axydx)/dT)-∫(d(Azydz)/dT)=vtUsy/c+∫(dUsy/(cdT))cdt-vxAxy-∫(dAxy/dT)dx-vzAzy-∫(dAzy/dT)dz=vtµ0∬(ρ0vt)(dy)2+µ0∫(d(∬(ρ0vt)(dy)2)/dT)cdt-vxµ0∬jx(dy)2-µ0∫(d(∬jx(dy)2)/dT)dx-vzµ0∬jz(dy)2-µ0∫(d(∬jz(dy)2)/dT)dz
Uz=∫Ezdz=∫(d(Uszcdt)/(cdT))-dΦxz/dT-dΦyz/dT=∫(d(Uszcdt)/(cdT))-∫(d(Axzdx)/dT)-∫(d(Ayzdy)/dT)=vtUsz/c+∫(dUsz/(cdT))cdt-vxAxz-∫(dAxz/dT)dx-vyAyz-∫(dAyz/dT)dy= =vtµ0∬(ρ0vt)(dz)2+µ0∫(d(∬(ρ0vt)(dz)2)/dT)cdt-vxµ0∬jx(dz)2-µ0∫(d(∬jx(dz)2)/dT)dx-vyµ0∬jy(dz)2-µ0∫(d(∬jy(dz)2)/dT)dy
Uct=∫Ectcdt=∫(d(Axctdx)/dT)+∫(d(Ayctdy)/dT)+∫(d(Azctdz)/dT)=vxAxct+∫(dAxct/dT)dx+vyAyct+∫(dAyct/dT)dy+vzAzct+∫(dAzct/dT)dz=vxµ0∬jx(cdt)2+µ0∫(d(∬jx(cdt)2)/dT)dx+vyµ0∬jy(cdt)2+µ0∫(d(∬jy(cdt)2)/dT)dy+vzµ0∬jz(cdt)2+µ0∫(d(∬jz(cdt)2)/dT)dz
U=Ux+Uy+Uz+Uct=∫Exdx+∫Eydy+∫Ezdz+∫Ectcdt=∫(d(Uscdt)/(cdT))-∫(d(Axdx)/dT)-∫(d(Aydy)/dT)-∫(d(Azdz)/dT)=vtUs/c+∫(dUs/(cdT))cdt-vxAx-∫(dAx/dT)dx-vyAy-∫(dAy/dT)dy-vzAz-∫(dAz/dT)dz=vtµ0∬(ρ0vt)((dx)2+(dy)2+(dz)2)+µ0∫(d(∬(ρ0vt)((dx)2+(dy)2+(dz)2))/dT)cdt-vxµ0∬jx((dy)2+(dz)2-(cdt)2-µ0∫(d(∬jx((dy)2+(dz)2-(cdt)2))/dT)dx-vyµ0∬jy((dx)2+(dz)2-(cdt)2-µ0∫(d(∬jy((dx)2+(dz)2-(cdt)2))/dT)dy-vzµ0∬jz((dx)2+(dy)2-(cdt)2-µ0∫(d(∬jz((dx)2+(dy)2-(cdt)2))/dT)dz
Where U is the electric potential , Ux is the electric potential in x-direction , Uy is the electric potential in y-direction , Uz is the electric potential in z-direction and Uct is the electric potential in the time dimension.
vx2+vy2+vz2+vt2=c2 c=(vx;vy;vz;vt)
vx is the x-komponent of the velocity , vy is the y-komponent of the velocity , vz is the z-komponent of the velocity and vt is the time velocity , c is the standard light speed (4velocity), ρ0 is the charge density and jx is the x-komponent of the current density , jy is the y-komponent of the current density , jz is the z-komponent of the current density , µ0 is the magnetical constant
jx2+jy2+jz2+(ρ0vt)2=(ρ0c)2 ρ0c=(jx;jy;jz;(ρ0vt))
(ds4)2=(cdT)2=(dx)2+(dy)2+(dz)2+(cdt)2
ds4=cdT=(dx;dy;dz;cdt)
Where ds4 is the smallest possible 4distance and dT is the smallest possible own time interval
E2=Ex2+Ey2+Ez2+Ect2 E=(Ex;Ey;Ez;Ect)
Where E is the electric field and Ex is the x-komponent of the electric field , Ey is the y-komponent of the electric field , Ez is the z-komponent of the electric field and Ect is the electric field komponent in the time dimension.
This supplement should be read together whit other parts of euclidean 4dimensional electromagnetism (for instance the formula for U is corrected in this supplement( that formula vas a little bit wrong in ”euclidean 4dimensional electromagnetism” however the formulas for Ux Uy Uz and Uct are correct in ”euclidean 4 dimensional electromagnetism”)) and is mostly about the vector potential and how it fits together whit the electric potential ( that is a scalar) I hope that this would help you to construct time (zero point) energy converters.
