# Electric field of an electric dipole

1. May 6, 2007

### Emanuel84

1. The problem statement, all variables and given/known data
Since the electrostatic field is conservative, show that it is irrotational for an electric dipole, whose dipole momentum is $$p$$.

2. Relevant equations
$$\nabla \times \mathbf{E} = 0$$

3. The attempt at a solution
I know that the components of the electric field in spherical coordinates are:

$$E_r = \frac{2 p \cos \theta}{4 \pi \epsilon_0 r^3}$$

$$E_\theta = \frac{p \sin \theta}{4 \pi \epsilon_0 r^3}$$

$$E_\phi = 0$$

so applying the curl is just a matter of calculus, and it's easy to show that
$$\nabla \times \mathbf{E} = 0$$.

Otherwise, using cartesian coordinates, if I choose the z-axis oriented as the dipole and set the origin in the dipole's center, the components of the electric field are:

$$E_x = \frac{p}{4 \pi \epsilon_0} \frac{3 x z}{r^5}$$

$$E_y = \frac{p}{4 \pi \epsilon_0} \frac{3 y z}{r^5}$$

$$E_z = \frac{p}{4 \pi \epsilon_0} \left( \frac{3z^2}{r^5} - \frac{1}{r^3} \right)$$

and the curl is different from 0, as one can easily prove, in contradiction with the previous result!

So, my question is:

Did I mistake or miss something? I really can't see what's wrong with this problem, at this time.. :uhh:

Thank you.

Last edited: May 6, 2007
2. May 18, 2007

### Emanuel84

Here is a quick computation I made with Mathematica regarding this problem.

As you can clearly see, in one case the curl is 0, in the second one is different from 0.

#### Attached Files:

• ###### Electric Field of An Electric Dipole.nb
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3. May 18, 2007

### Emanuel84

I finally realized Mathematica didn't do all the simplifications! :rofl:

By using Simplify command it comes up that curl(E)=(0,0,0) even in cartesian coordinates, as it should be.

Thank you, anyway!

Last edited: May 18, 2007