︠051eb4f0-570b-42cd-9c9a-706ab329cda7︠ # 计算积分，积分结果为磁场强度B = B1+B2 # pi = 3.14159 # u0 = 4*pi*10^(-7) # I = 10A # n = 2.5*10^5 # R1 = 0.1m # R2 = 0.3m # L = 0.2m # a = 0.1m var('u0 I n R1 R2 r a L') x = var('x') y = var('y') assume(a>0) assume(R1>0) assume(R2-R1>0) B = integral( integral(1/2*u0*I*n*y^2/(y^2+(r-x)^2)^(2/3),x,-a/2,a/2) , y, R1,R2) + integral( integral(1/2*u0*I*n*y^2/(y^2+(L-r+x)^2)^(2/3),x,-a/2,a/2) , y, R1,R2) show(B) ︡207721bf-70f1-4e79-a622-4d7216ffe9a4︡{"stdout": "
\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, I n u_{0} \\int_{R_{1}}^{R_{2}} y^{2} \\int_{-\\frac{1}{2} \\, a}^{\\frac{1}{2} \\, a} \\frac{1}{{\\left({\\left(L - r + x\\right)}^{2} + y^{2}\\right)}^{\\frac{2}{3"}︡ ︠615c0b0c-508d-412c-b381-b99cc6b51869i︠ %html ,{d x}\,{d y} + \frac{1}{2} \, I n u_{0} \int_{R_{1}}^{R_{2}} y^{2} \int_{-\frac{1}{2} \, a}^{\frac{1}{2} \, a} \frac{1}{{\left({\left(r - x\right)}^{2} + y^{2}\right)}^{\frac{2}{3}}}\,{d x}\,{d y}
}}} ︡6779cdc6-8ad2-4d36-bf0a-f278379adc47︡{"html": ",{d x}\\,{d y} + \\frac{1}{2} \\, I n u_{0} \\int_{R_{1}}^{R_{2}} y^{2} \\int_{-\\frac{1}{2} \\, a}^{\\frac{1}{2} \\, a} \\frac{1}{{\\left({\\left(r - x\\right)}^{2} + y^{2}\\right)}^{\\frac{2}{3}}}\\,{d x}\\,{d y}\n}}}"}︡ ︠c8035f56-362c-4aa1-80d9-2b7cb40653f5︠ # 计算积分，积分结果为磁场强度B = B1+B2 var('u0 I n R1 R2 r a L') t = var('t') assume(a>0) assume(R1>0) assume(R2-R1>0) print "B1为上面积分式中第一个二重积分." #B1对x积分，可以积分出结果：B1为上面积分式中第一个二重积分 print "B1对x的积分结果B11为：" B11=integral(-u0*I*n/2*y^2/(y^2+t^2)^(3/2),t,r-a/2,r+a/2) show(B11) #但B1对x积分后的结果B11对y积分，就无法积分出结果 print "B1对x积分后的结果B11对y积分，就无法积分出结果:" B12=integral(B11,y,R1,R2) show(B12) #如果对B11前半部分单独积分，仍不能积分出结果： print "如果对B11前半部分单独积分，仍不能积分出结果:" B111=(a-2*r)*sqrt(a^2-4*a*r+4*r^2+4*y^2) / (4*y^4+(a^2-4*a*r+4*r^2)*y^2) * y^2 show(integral(B111,y,R1,R2)) ︡d3abee57-2c97-4cc8-9da9-7d1663416af9︡{"stdout": "B1\u4e3a\u4e0a\u9762\u79ef\u5206\u5f0f\u4e2d\u7b2c\u4e00\u4e2a\u4e8c\u91cd\u79ef\u5206.\nB1\u5bf9x\u7684\u79ef\u5206\u7ed3\u679cB11\u4e3a\uff1a"}︡{"stdout": "B1\u4e3a\u4e0a\u9762\u79ef\u5206\u5f0f\u4e2d\u7b2c\u4e00\u4e2a\u4e8c\u91cd\u79ef\u5206.\nB1\u5bf9x\u7684\u79ef\u5206\u7ed3\u679cB11\u4e3a\uff1a\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(\\frac{{\\left(a - 2 \\, r\\right)} \\sqrt{a^{2} - 4 \\, a r + 4 \\, r^{2} + 4 \\, y^{2"}︡ ︠0c5c2423-8474-49fb-a4d8-bc10bb0881d7i︠ %html 4 \, y^{4} + {\left(a^{2} - 4 \, a r + 4 \, r^{2}\right)} y^{2}} + \frac{{\left(a + 2 \, r\right)} \sqrt{a^{2} + 4 \, a r + 4 \, r^{2} + 4 \, y^{2}}}{4 \, y^{4} + {\left(a^{2} + 4 \, a r + 4 \, r^{2}\right)} y^{2}}\right)} I n u_{0} y^{2}
B1对x积分后的结果B11对y积分，就无法积分出结果:
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, I n u_{0} \int_{R_{1}}^{R_{2}} {\left(\frac{{\left(a - 2 \, r\right)} \sqrt{a^{2} - 4 \, a r + 4 \, r^{2} + 4 \, y^{2}}}{4 \, y^{4} + {\left(a^{2} - 4 \, a r + 4 \, r^{2}\right)} y^{2}} + \frac{{\left(a + 2 \, r\right)} \sqrt{a^{2} + 4 \, a r + 4 \, r^{2} + 4 \, y^{2}}}{4 \, y^{4} + {\left(a^{2} + 4 \, a r + 4 \, r^{2}\right)} y^{2}}\right)} y^{2}\,{d y}
如果对B11前半部分单独积分，仍不能积分出结果:
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(a - 2 \, r\right)} \int_{R_{1}}^{R_{2}} \frac{\sqrt{a^{2} - 4 \, a r + 4 \, r^{2} + 4 \, y^{2}} y^{2}}{4 \, y^{4} + {\left(a^{2} - 4 \, a r + 4 \, r^{2}\right)} y^{2}}\,{d y}
}}} ︡352bfdd4-2313-49c2-bccc-9e1c85f2507b︡{"html": "4 \\, y^{4} + {\\left(a^{2} - 4 \\, a r + 4 \\, r^{2}\\right)} y^{2}} + \\frac{{\\left(a + 2 \\, r\\right)} \\sqrt{a^{2} + 4 \\, a r + 4 \\, r^{2} + 4 \\, y^{2}}}{4 \\, y^{4} + {\\left(a^{2} + 4 \\, a r + 4 \\, r^{2}\\right)} y^{2}}\\right)} I n u_{0} y^{2}\nB1\u5bf9x\u79ef\u5206\u540e\u7684\u7ed3\u679cB11\u5bf9y\u79ef\u5206\uff0c\u5c31\u65e0\u6cd5\u79ef\u5206\u51fa\u7ed3\u679c:\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, I n u_{0} \\int_{R_{1}}^{R_{2}} {\\left(\\frac{{\\left(a - 2 \\, r\\right)} \\sqrt{a^{2} - 4 \\, a r + 4 \\, r^{2} + 4 \\, y^{2}}}{4 \\, y^{4} + {\\left(a^{2} - 4 \\, a r + 4 \\, r^{2}\\right)} y^{2}} + \\frac{{\\left(a + 2 \\, r\\right)} \\sqrt{a^{2} + 4 \\, a r + 4 \\, r^{2} + 4 \\, y^{2}}}{4 \\, y^{4} + {\\left(a^{2} + 4 \\, a r + 4 \\, r^{2}\\right)} y^{2}}\\right)} y^{2}\\,{d y}
\n\u5982\u679c\u5bf9B11\u524d\u534a\u90e8\u5206\u5355\u72ec\u79ef\u5206\uff0c\u4ecd\u4e0d\u80fd\u79ef\u5206\u51fa\u7ed3\u679c:\n
\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(a - 2 \\, r\\right)} \\int_{R_{1}}^{R_{2}} \\frac{\\sqrt{a^{2} - 4 \\, a r + 4 \\, r^{2} + 4 \\, y^{2}} y^{2}}{4 \\, y^{4} + {\\left(a^{2} - 4 \\, a r + 4 \\, r^{2}\\right)} y^{2}}\\,{d y}
\n}}}"}︡ ︠eabd6763-c5ce-412d-8687-71fe144c5148︠ # 画出磁场B与a的关系 R1=3 R2=5 a=var('a') R = 4 n=4 u=2 I=1 #B=-2/3*(R1^3 - R2^3)*sqrt(4*R^2 + a^2)*I*a*n*u/(4*R^4 + R^2*a^2) #plot(B, (a,0,50),rgbcolor=hue(0.6)) ︡2bae7be0-9152-4ad0-bec5-2d5afc07ba3b︡︡ ︠8de00c01-a105-4681-8be1-8e9a54cb2421︠ # 画出磁场B与R的关系 R2=50 R1=30 n=4 u=2 b=2 R=var('R') a = 2 #B=-2/3*(R1^3 - R2^3)*sqrt(4*R^2 + a^2)*b*a*n*u/(4*R^4 + R^2*a^2) #plot(B, (R,30,50), rgbcolor=hue(0.3)) ︡cb4f390a-ab2c-4cc5-9d97-dce738fbc900︡︡ ︠be6be98c-4262-4101-a98f-1cfb3063ae13︠ # 画出磁场B与R和a的3d关系 R2=50 R1=30 n=4 u=2 b=2 R=var('R') a = var('a') #f=-2/3*(R1^3 - R2^3)*sqrt(4*R^2 + a^2)*b*a*n*u/(4*R^4 + R^2*a^2) #plot3d(f, (R,30,50), (a,0,50),rgbcolor=hue(0.8)) ︡87353fb4-c67f-4b24-b8aa-78344dce08cb︡︡