Torque on an arbitrary shaped object having some dipole moment in a uniform electric field

Question

Can anyone show me a proof that the torque on an arbitrary shaped object having a dipole moment $vec{P}$ in a uniform electric field $vec{E}$ is given by $mathbf{N}=mathbf{p} times mathbf{E}$.
I couldn't find it in my textbook or on the internet.

Yasir Sadiq · Answer

Assuming constant electric field we've
begin{align} 
vec{T} 
&=int vec{r} times(rho(vec{r}) vec{E} d tau) 
 &=int(vec{r} rho(vec{r}) d tau times E) 
 &=left(int vec{r} rho(vec{r}) d tauright) times vec{E} 
 &=vec{p} times vec{E} 
end{align}

Torque on an arbitrary shaped object having some dipole moment in a uniform electric field

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