Find the Electric field due to an electric dipole at the origin

[MatinSAR]

Homework Statement

Find electric field due to dipole at any point using $\vec E=-\nabla \phi$.

Relevant Equations

$\vec E=-\nabla \phi$

Question :

I have tried to solve but I struggle with this part:

Any help would be appreciated.

 

[kuruman]

Just expand the dot product and $\vec r$ then take derivatives.$$(\vec p\cdot \vec{\nabla})\vec r = \left(p_x\frac{\partial}{\partial x}+p_y\frac{\partial}{\partial y}+p_z\frac{\partial}{\partial z}\right)(x~\hat x+y~\hat y+z~\hat z)$$

 

[MatinSAR]

Just expand the dot product and $\vec r$ then take derivatives.$$(\vec p\cdot \vec{\nabla})\vec r = \left(p_x\frac{\partial}{\partial x}+p_y\frac{\partial}{\partial y}+p_z\frac{\partial}{\partial z}\right)(x~\hat x+y~\hat y+z~\hat z)$$

Then it is equal to $Pr^{-3}$, Am I right?!

 

[kuruman]

Show me the math.

 

[MatinSAR]

Show me the math.

Sorry that $r^{-3}$ was related to another part.

 

[MatinSAR]

I have used:
$p_x\frac{\partial}{\partial x}y=0$
$p_x\frac{\partial}{\partial x}z=0$
$p_y\frac{\partial}{\partial y}x=0$
$p_y\frac{\partial}{\partial y}z=0$
$p_z\frac{\partial}{\partial z}x=0$
$p_z\frac{\partial}{\partial z}y=0$

$p_x\frac{\partial}{\partial x}x=p_x$
$p_y\frac{\partial}{\partial y}y=p_y$
$p_z\frac{\partial}{\partial z}z=p_z$

 

[kuruman]

I have used:
$p_x\frac{\partial}{\partial x}y=0$
$p_x\frac{\partial}{\partial x}z=0$
$p_y\frac{\partial}{\partial y}x=0$
$p_y\frac{\partial}{\partial y}z=0$
$p_z\frac{\partial}{\partial z}x=0$
$p_z\frac{\partial}{\partial z}y=0$

$p_x\frac{\partial}{\partial x}x=p_x$
$p_y\frac{\partial}{\partial y}y=p_y$
$p_z\frac{\partial}{\partial z}z=p_z$

Yes. What is your final answer when you put it all together?

 

[MatinSAR]

Yes. What is your final answer when you put it all together?

$\vec P$ , I guess.

 

[kuruman]

Sorry, not that. I meant putting together the final expression $\vec E=-\vec{\nabla}\psi=?$

 

[MatinSAR]

Sorry, not that. I meant putting together the final expression $\vec E=-\vec{\nabla}\psi=?$

I am trying to solve ... I will send the work.

Thanks again for your help Prof.Kuruman.

 

[MatinSAR]

@kuruman
Final answer.

 

[kuruman]

That's it. Good job!

 

[MatinSAR]

That's it. Good job!

Thanks a lot! Have a good day.