 #1
 55
 2
Homework Statement:
 Using the vector potential A, show that the Cartesian representation of the magnetic induction field associated with a magnetic moment oriented along the Cartesian zaxis is B
Relevant Equations:

[tex]\vec{B} = \nabla \times \vec{A}[/tex]
[tex]\vec{A} = \frac{\mu}{4\pi}\frac{m_z}{r^3} (y,x,0)[/tex]
[tex]\vec{B} = \frac{\mu}{4\pi}\frac{m_z}{r^5} (3xz, 3yz, 3z^2r^2)[/tex]
[tex]\frac{\partial}{\partial x} \frac{1}{r^3} = \frac{3x}{r^5}[/tex]
So I was able to do out the curl in the i and j direction and got 3xz/r^{5} and 3yz/r^{5} as expected. However, when I do out the last curl, I do not get 3z^{2}3r^{2}. I get the following
[tex]\frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
[tex]\frac{\partial}{\partial y} \frac{y}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{2y^2+x^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
which when added together gives me
[tex]\frac{x^2y^2+2z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex].
I can't see where I've gone wrong with this differentiation. I've tried it out on symbolab and get the same result.
[tex]\frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
[tex]\frac{\partial}{\partial y} \frac{y}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{2y^2+x^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
which when added together gives me
[tex]\frac{x^2y^2+2z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex].
I can't see where I've gone wrong with this differentiation. I've tried it out on symbolab and get the same result.