# Divergence of an Electric Field due to an ideal dipole

## Homework Statement:

Am I tackling this problem in a correct way and if, is there any other way?

## Relevant Equations:

##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##
Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$

It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to show that ##\nabla \cdot \vec E = - \nabla^2 \phi = 0##.
My assumption of this equal ##0##, is from Gauss Law $$\iint_{\partial V} \vec E \cdot d\vec S = \iiint_V \nabla \cdot \vec E \, dV = \frac {Q_{encl}} {\epsilon_{0}}$$

However, when ##\vec d \rightarrow 0## we can't enclose one of the charges without enclosing the other one with any surface, hence ##\Sigma Q = 0##,
thus ##Q_{encl} = 0 \leftrightarrow \nabla \cdot \vec E = 0##

My questions: Am I reasoning correct, and does it exist any other way to show it? I'd rather not do a bruteforce calculation of ##- \nabla^2 \phi ##. =D

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
Clearly, as you have concluded, the divergence must be zero everywhere due to Gauss' law (at least away from the dipole, where things get more complicated than that - but that complication is most likely beyond the scope of your class and the problem formulation). The question becomes what you are allowed to assume and how strict you need to make your argument. For example, if you can start from the potential of each individual charge and the corresponding point-charge potential and use that to argue for the potential of the dipole, then that should be sufficient.

There is also nothing wrong with the brute force calculation, which is rather straightforward in this case given some basic identities regarding the derivatives of ##\vec r## and ##r##.

There is also nothing wrong with the brute force calculation, which is rather straightforward in this case given some basic identities regarding the derivatives of ##\vec r## and ##r##.
Through calculations, given ##\vec P## is constant, can you assign ##\vec P## the values ##\vec P = (P_{1}, P_{2}, P_{3})## and regard the components as constants?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
Through calculations, given ##\vec P## is constant, can you assign ##\vec P## the values ##\vec P = (P_{1}, P_{2}, P_{3})## and regard the components as constants?
Assuming you are writing the components in a Cartesian basis, fine. However, it should be sufficient to know that ##\vec p## is a constant vector, you do not really need to assign coordinates unless you need to derive the derivatives of the position vector ##\vec r## and the distance ##r##. Of course, you can assign a Cartesian basis if that makes you happier. The final result will not depend on the choice of basis.

• rakso
Assuming you are writing the components in a Cartesian basis, fine. However, it should be sufficient to know that ##\vec p## is a constant vector, you do not really need to assign coordinates unless you need to derive the derivatives of the position vector ##\vec r## and the distance ##r##. Of course, you can assign a Cartesian basis if that makes you happier. The final result will not depend on the choice of basis.
Im quite confused how to derive the dot product. Through the quotient rule I obtain $$\nabla \phi_{x} = \frac {(\partial_{x} (\vec P \cdot \vec r))r^3) - (\vec P \cdot \vec r)(\partial_{x}r^3)} {r^6}$$

Which, makes it complicated to execute the Lapace operator $$-\nabla^2 \phi = -\partial_{i} \partial_{i} \phi$$ for the three components. Am I missing something?

Orodruin
Staff Emeritus
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