# What is the complex dipole moment?

I have some problems with getting the complex (time dependent) dipole moments of some dipoles in a configuration. I eventually want to get the electric and magnetic fields of the configuration, but my biggest confusion is about the dipole moments. I wonder:

• How can you obtain the complex dipole moments mathematically?
• Does it mean anything physically when the dipole moment is complex? Or is it only used for simpler math?

I can give you an example:

Consider the following configuration. I want to calculate the complex dipole moments of the two dipoles (and later the electric field they are creating). The dipoles have the dipole moments

$\vec{p}_{1}=p_0cos(\omega t)\hat{z}$

$\vec{p}_{2}=p_0cos(\omega t-\pi/2)\hat{z}$

I would then use the Euler identity

$e^{ix}=cos(x)+isin(x)$

$cos(x)=\frac{e^{ix}+e^{-ix}}{2}$

$sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$

which would give me the following dipole moments

$\vec{p}_{1}=p_0\frac{e^{i\omega t}+e^{-i\omega t}}{2}\hat{z}$

$\vec{p}_{2}=p_0\frac{e^{i(\omega t-\pi/2)}-e^{-i(\omega t -\pi/2)}}{2}\hat{z}$

However, according to my book the correct complex dipole moments should be

$\vec{p}_{1}=p_0\hat{z}$

$\vec{p}_{2}=p_0e^{-i\pi/2}\hat{z}$

I can see that the author has removed the time dependent part, but I still think my result looks wrong. I feel quite confused about the complex dipole moments so I hope someone can make it a little bit clearer.

## 1 Answer

If you use the identity

$$\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha\sin\beta$$

You can see that for $\beta = \pi/2$ some terms will be zero. What you are left with can then be written in terms of $p_0$ (which itself it time varying).

You can actually get the same result by taking your expression and realizing that

$$\cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}$$

and

$$\sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2}$$

I agree that it's a little bit confusing, but the math works out. You were very close. Having a few trig identities at your fingertips (recognizing them in what is written down) really helps with problems such as these.

• Thank you! I can see how this works for the $\pi/2$, but I still don't understand how the time dependent terms disappeared. I would like to break the factor $e^{-i\omega t}$ out of the equation and then neglect it, but I don't see how this can be done with $(e^{i\omega t}+e^{-i\omega t})/2$. If we could just neglect the terms from the start, then we should have the extra factor $1/2$ in the answer? Also, do you know the answer to my second question? What is the physical meaning of complex dipole moment, or are we just rewriting things to make math simpler? – Djamillah Jan 5 '16 at 13:45
• The time dependent term is absorbed into $p_0$ I think. – Floris Jan 5 '16 at 13:57
• And complex numbers are typically just a way to condense time variations / phase shifts in a compact manner. – Floris Jan 5 '16 at 13:58