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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

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

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