25.3 The Potential Energy of a Dipole

The electric dipole has been our model for understanding how charged objects interact with neutral objects. In Chapter 23 we found that an electric field exerts a torque on a dipole. We can complete the picture by calculating the potential energy of an electric dipole in a uniform electric field.

FIGURE 25.13 shows a dipole in an electric field $\vec{E}$ . Recall that the dipole moment $\vec{p}$ is a vector that points from $- q$ to q with magnitude $p = q s$ . The forces $\vec{F}_{+}$ and $\vec{F}_{-}$ exert a torque on the dipole, but now we’re interested in calculating the work done by these forces as the dipole rotates from angle $ϕ_{i}$ to angle $ϕ_{f}$ .

The figure illustrates the work done by an electric field as the dipole rotates.

FIGURE 25.13 Full Alternative Text

When a force component $F_{s}$ acts through a small displacement $d s$ , the force does work $d W = F_{s}_{} d s$ . If we exploit the rotational-linear motion analogy from Chapter 12, where torque $τ$ is the analog of force and angular displacement $Δ ϕ$ is the analog of linear displacement, then a torque acting through a small angular displacement $d ϕ$ does work $d W = τ d ϕ$ . From Chapter 23, the torque on the dipole in Figure 25.13 is $τ = - p E sin ϕ$ , where the minus sign is due to the torque trying to cause a clockwise rotation. Thus the work done by the electric field on the dipole as it rotates through the small angle $d ϕ$ is

d W_{elec} = - p E sin ϕ d ϕ

(25.16)

The total work done by the electric field as the dipole turns from $ϕ_{i}$ to $ϕ_{f}$ is

W_{elec} = - p E \int_{ϕ_{i}}^{ϕ_{f}} sin ϕ d ϕ = p E cos ϕ_{f} - p E cos ϕ_{i}

(25.17)

The potential energy associated with the work done on the dipole is

Δ U_{elec} = U_{f} - U_{i} = - W_{elec} (i \to f) = - p E cos ϕ_{f} + p E cos ϕ_{i}

(25.18)

By comparing the left and right sides of Equation 25.18, we see that the potential energy of an electric dipole $\vec{p}$ in a uniform electric field $\vec{E}$ is

U_{elec} = - p E cos ϕ = - \vec{p} \cdot \vec{E}

(25.19)

FIGURE 25.14 shows the energy diagram of a dipole. The potential energy is minimum at $ϕ = 0 °$ where the dipole is aligned with the electric field. This is a point of stable equilibrium. A dipole exactly opposite $\vec{E}$ , at $ϕ = \pm 180 °$ , is at a point of unstable equilibrium. Any disturbance will cause it to flip around. A frictionless dipole with mechanical energy $E_{mech}$ will oscillate back and forth between turning points on either side of $ϕ = 0 °$ .

The figure illustrates the energy of a dipole in an electric field.

FIGURE 25.14 Full Alternative Text

EXAMPLE 25.5 Rotating a molecule

The water molecule is a permanent electric dipole with dipole moment $6.2 \times 10^{- 30} C m$ . A water molecule is aligned in an electric field with field strength $1.0 \times 10^{7} N / C$ . How much energy is needed to rotate the molecule $90 ° ?$

MODEL The molecule is at the point of minimum energy. It won’t spontaneously rotate $90 °$ . However, an external force that supplies energy, such as a collision with another molecule, can cause the water molecule to rotate.

SOLVE The molecule starts at $ϕ_{i} = 0 °$ and ends at $ϕ_{f} = 90 °$ . The increase in potential energy is

\begin{array}{l} Δ U_{elec} & = U_{f} - U_{i} = - p E \cos 90 ° - (- p E \cos 0 °) \\ = p E = 6.2 \times 10^{- 23} J \end{array}

This is the energy needed to rotate the molecule $90 °$ .

REVIEW $Δ U_{elec}$ is significantly less than $k_{B} T$ at room temperature. Thus collisions with other molecules can easily supply the energy to rotate the water molecules and keep them from staying aligned with the electric field.

25.3 The Potential Energy of a Dipole

FIGURE 25.13 The electric field does work as a dipole rotates.

FIGURE 25.14 The energy of a dipole in an electric field.

EXAMPLE 25.5 Rotating a molecule