1. Equipotentials and Electric Field lines
From a charged plate and Inside a capacitor
by Charles Young
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2. 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
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3. 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
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4. 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
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5. 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)
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6. Laplace’s Equation
Zero means there’s no charge at the boundary condition
Boundary condition 100 volts
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7. Taylor Expansion
By performing taylor expansion on φ(x+Δx,y) and φ(x-Δx,y) give us
And
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8. 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))
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9. Results
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10. 1000 Volts, grid of 60x60
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11. Switched signs of voltage
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12. 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
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13. Electric Field Lines of a Capacitor
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14. 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.
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15. 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
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