📂Electrodynamics

Alignment of Polar Molecules by a Constant External Electric Field

Overview ¹

When an electrically neutral atom is placed in an external electric field, it becomes polarized and acquires a dipole moment $\mathbf{p}$. However, some molecules have a dipole moment even without the influence of an external electric field. Such molecules are referred to as polar molecules.

Polar Molecules

An example of a polar molecule is a water molecule. Water molecules are bent as shown in $105^{\circ}$, resulting in a difference in polarity between the top and bottom parts, as illustrated in the figure above. Particularly because water has a large dipole moment, it serves as an effective solvent.

When the external electric field is constant, the sum of the forces on the positive and negative charges in a polar molecule is $\mathbf{0}$. Hence, while it might seem that the external electric field would have no effect on a polar molecule, as shown in the figure below, it experiences a torque (rotational force).

The torque experienced by a polar molecule can be calculated as follows. Let us denote the vector from the center of the dipole moment to $+q$ as $\frac{1}{2}\mathbf{d}$. The force experienced by a point charge $q$ due to the electric field $\mathbf{E}$ is $\mathbf{F}=q\mathbf{E}$, hence

$$\begin{align*} \mathbf{N} =&\ \left[ \frac{1}{2}\mathbf{d} \times \mathbf{F}_+ \right] + \left[ -\frac{1}{2}\mathbf{d} \times \mathbf{F}_- \right] \\ =&\ \left[ \frac{1}{2}\mathbf{d} \times (q\mathbf{E} ) \right]+ \left[ -\frac{1}{2}\mathbf{d} \times (-q\mathbf{E}) \right] \\ =&\ q\mathbf{d} \times \mathbf{E} \end{align*}$$

Therefore, a dipole $\mathbf{p} = q\mathbf{d}$ in a uniform electric field $\mathbf{E}$ experiences the following torque:

$$\mathbf{N} = \mathbf{p} \times \mathbf{E}$$

The torque $\mathbf{N}$ causes the dipole moment $\mathbf{p}$ to align parallel to the external electric field $\mathbf{E}$. In other words, while an external electric field induces a dipole moment in a neutral atom, in the case of polar molecules that already have a dipole moment, the dipole moment rotates to align itself with the external electric field.