Mawell's stress tensor

1. Mar 28, 2005

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

Last edited: Mar 28, 2005
2. Mar 28, 2005

Dathascome

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

3. Mar 28, 2005

StatusX

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

4. Mar 28, 2005

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