Understanding the Force on a Charged Hemisphere Using the Stress Tensor

[Dathascome]

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 $$(\vec(T) \cdot \vec(da))_z$$( sorry I don't know how to right T as a tensor and not a vector). In the book they get $$(\vec(T) \cdot \vec(da))_z=\epsilon_o/2\((Q/4\pi\epsilon_0R)^2\sin(\theta)cos(\theta)d\theta\ d\phi$$

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

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

 

[Dathascome]

Just wanted to bump this back up...I messed up and hit submit before finishing. I hope someone really reads it this time

 

[StatusX]

Are you sure you copied the tensor components correctly? In my book there is only an $\epsilon_0/2$ in front of T_zz. The other components just have an $\epsilon_0$.

 

[Dathascome]

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