Electric Dipole Due to Point Charges

Electric Dipole Due to Point Charges

An electric dipole is defined as a separation of charges. The simplest example is a pair of equal and opposite charges separated by a distance. However, a charge distribution, either volume, surface or line can also be approximated as an electric dipole from a large distance. These dipoles are characterized by their dipole moment, a vector quantity defined as the charge multiplied by their separation. The direction of the dipole moment vector is from the negative charge to the positive charge.
electric dipole

A real-life example of an electric dipole is a water molecule. A water molecule forms an angle, with hydrogen atoms at the tips and oxygen at the vertex. Since oxygen has a higher electronegativity than hydrogen, the side of the molecule with the oxygen atom has a partial negative charge. The charge differences cause water molecules to be attracted to each other and to other polar molecules. This attraction is known as hydrogen bonding. The dipolar property also makes water molecule a very good solvent.

water molecule as dipole

Electrostatic Potential Due to an Electric Dipole

In the following section we will calculate electrostatic potential due to an electric dipole. The dipole is shown in the diagram below. It consists of two equal and opposite charges separated by a distance d. The charge center is at point O. First we will find the exact potential at a test point P and we will also figure out the approximate potential at that point when it is far from the dipole charge center.
simplest electric dipole
Using superposition principle, the electrostatic potential due to this electric dipole at point P is

(1)   \begin{equation*} V(\vec{r})=\frac{1}{4\pi\epsilon_0}\left[\frac{q}{r_+}-\frac{q}{r_-}\right] \end{equation*}

This is the exact potential due to the dipole system. Here r_{+} and r_{-} are the distances of the test point P from the +q and -q charges, respectively. Now, we will express these distances in terms of the distance r which is calculated from the charge center. Using laws of vectors, we can easily write

    \[ r_{\pm}^2=\left[r^2+\left(\frac{d}{2}\right)^2 \mp rdcos\theta \right] \]

    \[ =r^2\left[ 1+\frac{1}{4}\left(\frac{d^2}{r^2}\right) \mp \frac{d}{r}cos\theta \right] \]

As already mentioned, we are interested in the region r>>d. Consequently, we can neglect higher order terms of \frac{d}{r}. Thus we can approximately write

    \[ r_{\pm}^2 \cong r^2\left[ 1 \mp \frac{d}{r}cos\theta \right] \]

    \[ \frac{1}{r_{\pm}} \cong \frac{1}{r}\left[ 1 \pm \frac{d}{2r}cos\theta \right] \]

Therefore we can write

    \[ \frac{1}{r_{+}}-\frac{1}{r_{-}} \cong \frac{1}{r}\left[ 1 + \frac{d}{2r}cos\theta \right]- \frac{1}{r}\left[ 1 - \frac{d}{2r}cos\theta \right] \]

    \[ \cong \left(\frac{d}{r^2}\right)cos\theta \]

    \[V(\vec{r})=\frac{1}{4\pi\epsilon_0}\left[\frac{qdcos\theta}{r^2}\right]\]

(2)   \begin{equation*} \boxed{V(\vec{r})=\frac{1}{4\pi\epsilon_0}\left[\frac{pcos\theta}{r^2}\right]} \end{equation*}

where p is the magnitude of the dipole moment. Please notice that the electrostatic potential is inversely proportional to the square of distance from the charge center.

Dipole potential polar coordinate
In the diagram above, we have shown the electrostatic potential in the plane polar coordinate due to an electric dipole. A simplified version of the above diagram can be plotted using gnuplot. The code is available here.

Electric Field Due to an Electric Dipole

We can easily figure out the electric field due to this dipole by calculating the negative gradient of the electrostatic potential. In spherical polar coordinate electric field will be independent of \phi coordinate.

    \[\frac{\partial V}{\partial r}=-\frac{1}{4\pi\epsilon_0}\left[ \frac{2qd cos\theta}{r^3} \right] \]

    \[ \frac{1}{r} \frac{\partial V}{\partial \theta}=-\frac{1}{4\pi\epsilon_0}\left[ \frac{qd sin\theta}{r^3} \right] \]

(3)   \begin{equation*} \boxed{\vec{E}=\frac{qd}{4\pi\epsilon_0}\left[\frac{2cos\theta}{r^3}\hat{r}+ \frac{sin\theta}{r^3}\hat{\theta} \right]} \end{equation*}

In contrast with an electric field produced by a point charge, electric field due to a dipole is proportional to r^{-3}. The electric field does not point along radius vector, however, it has both components along \hat{r} and \hat{\theta}.

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