How Do Two Dipoles Affect Each Other's Torque?

[minimark1234]

Homework Statement

Two dipoles as in pic below a distance r apart, find the torque applied to each dipole due to the other.
$$\uparrow1 ------------2\rightarrow$$

Homework Equations

$$\itshape \vec{p}=p(cos\:\theta \hat{r}-sin\:\theta\hat{\theta})$$
$$\itshape \vec{E}=\frac{p}{4\pi\epsilon_{0}r^{3}}\: (2cos\:\theta\hat{r}+sin\:\theta\hat{\theta})$$
$$\itshape \vec{N}=\vec{p}\:X\:\vec{E}$$

The Attempt at a Solution

well i only am concerning myself with the non constants in the cross product.
$$\itshape \vec{N}=\vec{p_{2}}\:X\:\vec{E_{1}}$$
$$\itshape (cos\:\theta\hat{r}-sin\:\theta\hat{\theta})X(2cos\:\theta\hat{r}+sin\:\theta\hat{\theta})$$
ok this is where i am confused... do i use \theta as angle under the z axis for the direction the dipole and electric field are pointing, or do i use them for angles of separation or what? if i use them for angle under the z axis for the vector it is pertaining to i get the torque will be $$-3cos\:\theta sin\:\theta sin \:\phi\hat{i}+3cos\:\theta sin\:\theta cos \:\phi\hat{j}$$, which applying angles under z axis is 0. But that isn't what the answer should be.

 

[Jhero]

Hi there, thank you for your question. In order to find the torque applied to each dipole due to the other, you will need to use the equations you have provided and consider the orientation of the dipoles and the distance between them. The angle theta should represent the angle between the dipole moment and the electric field, and the angle phi should represent the angle between the two dipoles. The torque will depend on the specific orientation and distance of the dipoles, so make sure to carefully consider these factors when solving for the torque. Also, remember to use the correct units and constants in your calculations. I hope this helps, let me know if you have any further questions. Good luck!