Equipotentials and Electric Field lines From a charged plate and Inside a capacitor by Charles Young Hirophysics.com

Mapping Equipotentials Compiled using Fortran (g77) No input data needed with program Plotted in gnuplot of Linux® ubuntu 9.04 Equipotentials are lines where potential energy is uniform, i.e. a charge moves along them without changing potential energy Can be measured and mapped by voltmeters unlike electric field Hirophysics.com

Procedure to Compile After writing the program, you must run it in the terminal by g77 laplace.f Then you must run output in the terminal ./a.out Then you must open gnuplot by running in terminal Gnuplot To plot the output data you must write Splot ‘laplace.d’ w l This means it’s a 3D plot with lines Hirophysics.com

Program Excerpt real*8 x, p(40,40) responsible for size of grid integer i, j, y, iter open(8, file='laplace.d', status='unknown') max = 40 do i=1, max do j=1, max p(i,j) = 0.0 initializing variables at 0 end do end do do i=1, max p(i,1)=100.0 responsible for voltage of plate end do Hirophysics.com

Iteration Algorithm do iter=1, 1000 do i=2, (max-1) do j=2, (max-1) p(i,j)=0.25*(p(i+1,j)+p(i-1,j)+p(i,j+1)+p(i,j-1)) end do end do end do Saving and formatting do i=1, max do j=1, max write(8,22) p(i,j) end do write(8,22) end do 22 format(f10.6) Hirophysics.com

Laplace’s Equation Zero means there’s no charge at the boundary condition Boundary condition 100 volts Hirophysics.com

Taylor Expansion By performing taylor expansion on φ(x+Δx,y) and φ(x-Δx,y) give us And Hirophysics.com

Substituting Taylor into Laplace Δx=Δy=Δ because we want the grid spaces to be equal Solve for φ(x,y) to get: Iterated algorithm in program: p(i,j)=0.25*(p(i+1,j)+p(i-1,j)+p(i,j+1)+p(i,j-1)) Hirophysics.com

Results Hirophysics.com

1000 Volts, grid of 60x60 Hirophysics.com

Switched signs of voltage Hirophysics.com

Electric Field lines Electric Field-electric force per unit charge at a point. Field lines describe the direction of the force E = σ/eo, where sigma= charge density Two plates, add vectors from each plate Hirophysics.com

Electric Field Lines of a Capacitor Hirophysics.com

Electric Field lines, two plates It seems much more difficult to map electric field lines between two plates than mapping equipotential lines from a sheet of charge. This part was not as successful. Hirophysics.com

Applications Electricity and Magnetism Allows an easy visualization for characteristics of charges around plates or in between two plates Electronic musical instruments: Theremin Designing high voltage equipment Sterilization of foods: milk and juice, patented 4/24/09 Pulsing electric field Assists in capillary liquid chromatography Hirophysics.com