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POTENTIAL IN A SEMICIRCLE ( 22 sept 2006)

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POTENTIAL IN A SEMICIRCLE ( 22 sept 2006)A,in,a,Sept

    POTENTIAL IN A SEMICIRCLE ( 22 sept 2006) by Reinaldo Baretti Machín

www.geocities.com/serienumerica

    e-mail: reibaretti2004@yahoo.com

    Analytical and numerical methods are employed to obtain the solution of Laplace equation in a semicircular region.In fig 1 the region is shown with a grounded base V(r,0 ) =V(r,π)=0. The

    circular region is kept at a constant V(r=a, φ ) =V . 0

    Fig. 1

In polar coordinates the Laplacian operating on the potential V(r,φ),

    has the form

    22222?V/?r (1/r) ?V/?r (1/r)? V /? φ =0 . (1)

The general solution is

    ? m-mV(r, φ)=? (A r +B r ) m=1mm

     (C sin(m φ) +D cos(m φ ) ). (2) mm

    -mSince the potential is finite at the origin the terms B r have to m

    be excluded.The potential is symmetric about the +Y axis i.e. V(r, φ) = V(r, π- φ)

and this excludes the terms D cos(m φ ), because they change m

    signs.

Our potential expression reduces to

    ? mV(r, φ)=? C r sin(m φ) (3) m=1m

    where C is a new coefficient to be found from the boundary m

    conditions.

At the boundary of the circle

    ? mV(r=a , φ ) = V =? C a sin(m φ) (4) 0m=1m

multiyplying by sin (n φ) and integrating from 0 to π we find the

    n-th coefficient . When integrating recall that the sine functions are

    orthogonal, so from the sum in the RHS only the n-th term remains

    n2? V sin (n φ) d φ = ? C a sin (n φ) d φ . (5) 0n

    If n is even sin (n φ) has a complete number of cycles and the integral on the left is zero.

    For odd n

    nV (2/n) = Ca (π/2) yields 0n

    n C= V (4/(n π ) (1/ a ) ( n-odd) . (6) n 0

    Substituting in (3) we have the final expression

    ? nnV(r, φ)=? ( V (4/(n π ) (1/ a ) ) r sin(nφ) (7) n=10

where the sum is over odd n .

A FORTRAN CODE is provided below that carries the summation.

    x,y,v= 0.000E+00 0.000E+00 -1.#JE+000

     x,y,v= 0.100E+00 0.000E+00 0.000E+00

     x,y,v= 0.200E+00 0.000E+00 0.000E+00

     x,y,v= 0.300E+00 0.000E+00 0.000E+00

     x,y,v= 0.400E+00 0.000E+00 0.000E+00

     x,y,v= 0.500E+00 0.000E+00 0.000E+00

     x,y,v= 0.600E+00 0.000E+00 0.000E+00

     x,y,v= 0.700E+00 0.000E+00 0.000E+00

     x,y,v= 0.800E+00 0.000E+00 0.000E+00

     x,y,v= 0.900E+00 0.000E+00 0.000E+00

     x,y,v= 0.100E+01 0.000E+00 0.000E+00

     x,y,v= 0.000E+00 0.100E+00 0.127E+02

     x,y,v= 0.100E+00 0.100E+00 0.128E+02

     x,y,v= 0.200E+00 0.100E+00 0.132E+02

     x,y,v= 0.300E+00 0.100E+00 0.139E+02

     x,y,v= 0.400E+00 0.100E+00 0.151E+02

     x,y,v= 0.500E+00 0.100E+00 0.168E+02

     x,y,v= 0.600E+00 0.100E+00 0.196E+02

     x,y,v= 0.700E+00 0.100E+00 0.242E+02

     x,y,v= 0.800E+00 0.100E+00 0.331E+02

     x,y,v= 0.900E+00 0.100E+00 0.538E+02

     x,y,v= 0.000E+00 0.200E+00 0.251E+02

     x,y,v= 0.100E+00 0.200E+00 0.254E+02

     x,y,v= 0.200E+00 0.200E+00 0.261E+02

     x,y,v= 0.300E+00 0.200E+00 0.274E+02

     x,y,v= 0.400E+00 0.200E+00 0.295E+02

     x,y,v= 0.500E+00 0.200E+00 0.327E+02

     x,y,v= 0.600E+00 0.200E+00 0.374E+02

     x,y,v= 0.700E+00 0.200E+00 0.449E+02

     x,y,v= 0.800E+00 0.200E+00 0.570E+02

     x,y,v= 0.900E+00 0.200E+00 0.766E+02

     x,y,v= 0.000E+00 0.300E+00 0.371E+02

     x,y,v= 0.100E+00 0.300E+00 0.374E+02

     x,y,v= 0.200E+00 0.300E+00 0.384E+02

     x,y,v= 0.300E+00 0.300E+00 0.402E+02

     x,y,v= 0.400E+00 0.300E+00 0.430E+02

     x,y,v= 0.500E+00 0.300E+00 0.470E+02

     x,y,v= 0.600E+00 0.300E+00 0.528E+02

     x,y,v= 0.700E+00 0.300E+00 0.611E+02

     x,y,v= 0.800E+00 0.300E+00 0.731E+02

     x,y,v= 0.900E+00 0.300E+00 0.904E+02

     x,y,v= 0.000E+00 0.400E+00 0.484E+02

     x,y,v= 0.100E+00 0.400E+00 0.488E+02

     x,y,v= 0.200E+00 0.400E+00 0.500E+02

     x,y,v= 0.300E+00 0.400E+00 0.521E+02

     x,y,v= 0.400E+00 0.400E+00 0.552E+02

     x,y,v= 0.500E+00 0.400E+00 0.595E+02

     x,y,v= 0.600E+00 0.400E+00 0.656E+02

     x,y,v= 0.700E+00 0.400E+00 0.737E+02

     x,y,v= 0.800E+00 0.400E+00 0.845E+02

     x,y,v= 0.900E+00 0.400E+00 0.954E+02

     x,y,v= 0.000E+00 0.500E+00 0.590E+02

     x,y,v= 0.100E+00 0.500E+00 0.594E+02

     x,y,v= 0.200E+00 0.500E+00 0.607E+02

     x,y,v= 0.300E+00 0.500E+00 0.629E+02

     x,y,v= 0.400E+00 0.500E+00 0.661E+02

     x,y,v= 0.500E+00 0.500E+00 0.705E+02

     x,y,v= 0.600E+00 0.500E+00 0.763E+02

     x,y,v= 0.700E+00 0.500E+00 0.838E+02

     x,y,v= 0.800E+00 0.500E+00 0.928E+02

     x,y,v= 0.000E+00 0.600E+00 0.688E+02

     x,y,v= 0.100E+00 0.600E+00 0.692E+02

     x,y,v= 0.200E+00 0.600E+00 0.705E+02

     x,y,v= 0.300E+00 0.600E+00 0.726E+02

     x,y,v= 0.400E+00 0.600E+00 0.758E+02

     x,y,v= 0.500E+00 0.600E+00 0.800E+02

     x,y,v= 0.600E+00 0.600E+00 0.854E+02

     x,y,v= 0.700E+00 0.600E+00 0.923E+02

     x,y,v= 0.800E+00 0.600E+00 0.100E+03

     x,y,v= 0.000E+00 0.700E+00 0.778E+02

     x,y,v= 0.100E+00 0.700E+00 0.782E+02

     x,y,v= 0.200E+00 0.700E+00 0.794E+02

     x,y,v= 0.300E+00 0.700E+00 0.814E+02

     x,y,v= 0.400E+00 0.700E+00 0.844E+02

     x,y,v= 0.500E+00 0.700E+00 0.883E+02

     x,y,v= 0.600E+00 0.700E+00 0.933E+02

     x,y,v= 0.700E+00 0.700E+00 0.971E+02

     x,y,v= 0.000E+00 0.800E+00 0.859E+02

     x,y,v= 0.100E+00 0.800E+00 0.863E+02

     x,y,v= 0.200E+00 0.800E+00 0.874E+02

     x,y,v= 0.300E+00 0.800E+00 0.894E+02

     x,y,v= 0.400E+00 0.800E+00 0.921E+02

     x,y,v= 0.500E+00 0.800E+00 0.954E+02

     x,y,v= 0.600E+00 0.800E+00 0.100E+03

     x,y,v= 0.000E+00 0.900E+00 0.932E+02

     x,y,v= 0.100E+00 0.900E+00 0.937E+02

     x,y,v= 0.200E+00 0.900E+00 0.946E+02

     x,y,v= 0.300E+00 0.900E+00 0.967E+02

     x,y,v= 0.400E+00 0.900E+00 0.980E+02

     x,y,v= 0.000E+00 0.100E+01 0.980E+02

     c potential of semicircular plate

     dimension v(0:50,0:50)

     V0=100.

     a=1.

     pi=2.*asin(1.)

     fact=4.*v0/pi

     nx=10

     ny=nx

     dx=1./float(nx)

     dy=1./float(ny)

     do 10 iy=0,ny

     do 10 ix=0,nx

     x=dx*float(ix)

     y=dy*float(iy)

     r=sqrt(x**2+y**2)

     phi=atan(y/x)

     if(r.le.a)then

     sum=0.

     do 20 n=1,31,2

     sum=sum+ (fact/(float(n)*a**n))*r**n*sin(float(n)*phi)

    20 continue

     v(ix,iy)=sum

     endif

    10 continue

     do 30 iy=0,ny

     do 30 ix=0,nx

     x=dx*float(ix)

     y=dy*float(iy)

     r=sqrt(x**2+y**2)

     if(r.le.a)then

     print 100,x,y, v(ix,iy)

     endif

    30 continue

    100 format(1x,'x,y,v=',3(4x,e10.3))

     stop

     end

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