**Electric dipole moment**

A pair of adjacent positive and negative charges is called an electric dipole, and an electric field is created around it.
In constant, most of materials are electrical neutral.
However, the charge of materials is strictly deviated at the molecular level.
This charge deviation generally affects the viscous and electro-magnetic characteristics of materials.

An electric dipole generates an electric field but also affects by the electric field applied by the external field.
The electric dipole has an electric dipole moment.
As it is difficult to understand what the electric dipole moment is, we carefully describe it below.

Figure 1. The adjacent electric charge

Here, we consider that the electric charges are set as \( (a,0,0) \) and \( (-a,0,0) \), as shown in Figure 1.
Note that the horizontal axis is the \(y\) axis and the vertical axis is the \(x\) axis.
These electric charges create the following electric field:
\begin{eqnarray}
E_r &=& \frac{2p \cos \theta}{4 \pi \epsilon_0 r^3} \ \ \ \ \ \ \ \ \ \ \ \ \ (1) \\
E_{\theta} &=& \frac{p \sin \theta}{4 \pi \epsilon_0 r^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
\end{eqnarray}
Equations (1) and (2) are expressed with polar coordinates.
These electric fields are derived under the assumption that the distance between the electric charges (2a) is sufficiently large \((r>>a)\).
\(p\) is expressed as \(p=2aQ\) and called the electric dipole moment.
The electric dipole moment can be expressed as a vector as follows:
\begin{eqnarray}
{\bf p} = Q{\bf d} \ \ \ \ \ \ \ \ \ (3)
\end{eqnarray}
where the vector \( {\bf d}\) is the distance vector and its scale is \({\bf p}=Q{\bf d}=2Qa\). Figure 2 represents the electric field created by the electric dipole moment.

Figure 2. The electric field generated by the electric dipole moment

Then, we consider the situation where the electric dipole is placed in a uniform electric field, as illustrated in Figure 3.

Figure 3. The electric dipole moment and electric field

In this case, the positive charge is the force \( F_+ = Q{\bf E} \) and the negative charge is the force \( F_- = -Q{\bf E}\).
Of course, as the electric charge \(Q\) is the same, \(|F_+|\) and \(|F_-|\) are equal and their direction is opposite.
Forces whose direction is opposite/parallel and equal size are called force couple.
When considered as an electric dipole which is a pair of electric charges, the resultant force applied to the electric dipole is zero.

Figure 4. The force of the electric field received from the electric field

If a pair of charges is placed at an angle \(\theta\) with the electric field, the force applied to each charge is parallel but not aligned on a straight line.
In this case the electric dipole rotates.
Then, the moment of force couple can be calculated.
The moment is calculated with the cross product between the position vector \( {\bf r} \) and force vector \( {\bf F} \).

We calculate the moment of positive and negative electric charges \(N_+\), \(N_-\), as follows:
\begin{eqnarray}
{\bf N}_+ &=& {\bf r}_+ \times {\bf F}_+ \ \ \ \ \ \ \ \ \ \ (4) \\
{\bf N}_- &=& {\bf r}_- \times {\bf F}_- \ \ \ \ \ \ \ \ \ \ (5)
\end{eqnarray}
As the couple moment can be expressed as \(N_+ + N_-\), we obtain:
\begin{eqnarray}
{\bf N} &=& {\bf N}_+ + {\bf N}_- \\
&=& Q \left( {\bf r}_+ - {\bf r}_- \right) \times {\bf E} \\
&=& Q {\bf d} \times {\bf E} \\
&=& {\bf p} \times {\bf E} \ \ \ \ \ \ \ \ \ \ \ \ \ (6)
\end{eqnarray}
The couple moment can be expressed using the electric dipole moment.

Figure 5. The moment of a couple

This couple moment reduces the angle \(\theta\) between the electric field and the electric dipole moment, as illustrated in Figure 5.

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