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Jackson 5.10 topic problem

  • Thread starter dingo_d
  • Start date
  • #1
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Homework Statement



There is one thing I'm not getting in this problem. It's about uniformly magnetized sphere. In solving the problem with formula for magnetic potential:

[tex]\phi_M=-\nabla\cdot\int\frac{\vec{M}}{|\vec{r}-\vec{r}'|}d^3r'[/tex]

At one point a change of variable for differentiation is made

[tex]\frac{\partial}{\partial z}=\frac{\partial r}{\partial z}\frac{\partial}{\partial r}[/tex]

And he says that [tex]\frac{\partial r}{\partial z}=\cos\theta[/tex]. But I can't see that. If I'm using the formula for spherical coordinate system transformation: [tex]z=r\cos\theta[/tex] I don't get just cosine, I get 1/cosine :\

So what formula is he using?

Thanks
 

Answers and Replies

  • #2
17
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He is using [itex] r = \sqrt {x^2 + y^2+z^2} [/itex], and then
[itex]
\frac{\partial r}{\partial z} = \frac{z}{r} = cos{(\theta)}
[/itex]
 
  • #3
fzero
Science Advisor
Homework Helper
Gold Member
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Since [tex] r = \sqrt{ x^2 + y^2 + z^2}[/tex],

[tex] \frac{\partial r }{\partial z} = \frac{z}{r}, [/tex]

which leads to the formula from the text. I think you're making a mistake because the partial derivatives in spherical and Cartesian coordinates are related by a 3x3 matrix, so it's not enough to merely compare reciprocals.
 
  • #4
211
0
oh! I see, thanks :D
 

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