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Mawell's stress tensor

  • Thread starter Dathascome
  • Start date
55
0
I'm having some trouble with an example in griffiths book about using the stress tensor. The problem is to find the force on the northern hemisphere of a uniformly charged solid sphere by the southern hemisphere. Charge Q, radius R. I understand that we will only need the zx, zy, and zz components of the tensor, and I can get those without a problem. The problem I have is with taking [tex](\vec(T) \cdot \vec(da))_z[/tex]( sorry I don't know how to right T as a tensor and not a vector). In the book they get [tex](\vec(T) \cdot \vec(da))_z=\epsilon_o/2\((Q/4\pi\epsilon_0R)^2\sin(\theta)cos(\theta)d\theta\ d\phi[/tex]

Where as I'm getting a cos ^3 instead of just a cos, and I can't see why.
I know that [tex]da=R^2sin(\theta)d\theta d\phi \hat{r}[/tex]
where [tex]\hat{r}=sin(\theta)cos(\phi)\hat{x}+sin(\theta)sin(\phi)\hat{y}+cos(\theta)\hat{z}[/tex]and that [tex]\epsilon_o/2\((Q/4\pi\epsilon_0R)^2[/tex]
along with
[tex]\vec(T)_z_x=\epsilon_o/2\((Q/4\pi\epsilon_0R)^2sin(\theta)cos(\theta)cos(\phi)[/tex]
[tex]\vec(T)_z_y=\epsilon_o/2\((Q/4\pi\epsilon_0R)^2sin(\theta)cos(\theta)sin(\phi)[/tex]
[tex]\vec(T)_z_y=\epsilon_o/2\((Q/4\pi\epsilon_0R)^2(cos(\theta)^2+sin(\theta)^2)[/tex]

So I take the dot product of each T_zx with da_x and the same with the other components but I don't cos, I get cos^3 for some reason.
Any help would be greatly appreciated...hope this wasn't too confusing o:)
 
Last edited:

Answers and Replies

55
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Just wanted to bump this back up...I messed up and hit submit before finishing. I hope someone really reads it this time :frown:
 
StatusX
Homework Helper
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Are you sure you copied the tensor components correctly? In my book there is only an [itex]\epsilon_0/2[/itex] in front of Tzz. The other components just have an [itex]\epsilon_0[/itex].
 
55
0
Doh...I think I did copy it wrong...let me do it over and see what happens.
Usually it's the first thing I check...I hate making stupid mistakes like that :mad:
 

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