Question

我正在尝试计算均匀充电的球形壳的电势和电场，并使用MATLAB在壳外部的空间中绘制结果。我已经尝试了从for循环到命令求和的所有操作，甚至我什至无法获得双积分的正确表达式（因为我可以计算出我的结果给出的结果与解析解不匹配）。

根据我的研究，最好的绘制方式是在同一张图上绘制不同等值的等值面图，以显示电势和电场如何随距球体的距离而变化。但这是我的第二个问题。我什至无法正确计算双积分。所以，我的问题是双重的，

1）如何使用包含作为常数的符号变量的黎曼和来计算双积分？我故意避免使用双积分命令（例如integral2（）或Riemann命令，例如trapz（）

）

2）考虑到我将具有三个变量的函数，如何在3维上绘制结果表达式？

我的代码的当前状态以计算电势（我先定义该状态，然后再使用梯度获取电场），如下所示：

syms r phi theta % symbolic values representing a random point in space 
% outside a uniformly charged spherical shell

% constants
N = 3; % total number of increments to divide the riemann sum into
Q = +20 .* 1e-9; % total charge on sphere
R = 1; % radius of sphere is 1 meter
sigma = Q/(4*pi*(R^2)); % uniform surface charge denisty
eps0 = 8.854e-12;
kC = 1/(4*pi*eps0); % coulomb's constant
dphi = 2*pi/N; % discretizing the interval over which I sum
dtheta = pi/N;

% Performing two riemann sums
% dA = R^2.*cos(thetaprime).*dphi.*dtheta; small area on surface of 
% sphere in spherical coordinates
phiprime = linspace(0,2*pi,N);
thetaprime = linspace(0,pi,N);
dInt1 = sym(zeros(0,N)); % preallocating space allowing the presence of 
symbols
dInt2 = sym(zeros(0,N));

for e = 1: length(phiprime)
for m = 1: length(thetaprime)
  dInt1(m) = dInt1 + 
((R^2).*cos(thetaprime).*dphi.*dtheta)./sqrt((r.*sin(theta).*cos(phi) - 
R.*sin(thetaprime).*cos(phiprime)).^2 + (r.*sin(theta).*sin(phi) - 
R.*sin(thetaprime).*sin(phiprime)).^2 + (r.*cos(theta) - 
R.*cos(phiprime)).^2)
end    

  dInt2(e) = dInt2 + sum(dInt1);

end
Int = sum(Int2);

Vtot = kC*sigma.*Int; % total symbolic expression for electric potential

%% Computing Values for Vtot at various points in space and plotting them

rinterval = linspace(1,4,12);
phiinterval = linspace (0,2*pi,12);
thetainterval = linspace(0,pi,12);
[X,Y,Z] = meshgrid(rinterval,phiinterval,thetainterval);

Vcont = subs(Vtot, [r,phi,theta], {X,Y,Z}); % substituting 2D coordinates 
now for symbolic values

对于此代码，我得到了错误：

Error using symengine
Array sizes must match.

Error in sym/privBinaryOp (line 1022)
        Csym = mupadmex(op,args{1}.s, args{2}.s, varargin{:});

Error in  +  (line 7)
X = privBinaryOp(A, B, 'symobj::zipWithImplicitExpansion', 
'_plus');

Error in Numerical_Integration_of_2D_Surfaces (line 27)
     dInt1(m) = dInt1 +    

((R^2).*cos(thetaprime).*dphi.*dtheta)./sqrt((r.*sin(theta).*cos(phi) -
     R.*sin(thetaprime).*cos(phiprime)).^2 + (r.*sin(theta).*sin(phi) -
     R.*sin(thetaprime).*sin(phiprime)).^2 + (r.*cos(theta) - 
     R.*cos(phiprime)).^2)

使用两个riemann和计算双积分并绘制多个等值面

0 个答案: