PX157 - B3 - electrical dipoles

a pair of charges, $+ q$ and $- q$ , separated by a fixed distance, $d$
the electrical dipole moment is:

\vec{p} = q \vec{d}

in an external electric field, $\vec{E}$ , force on the dipole is:

\vec{F} = + q \vec{E} + (- q) \vec{E} = 0

total force is zero, but there is a torque
angular momentum: $\vec{L} = \vec{r} \times m \vec{v}$
angular force (torque): $\vec{τ} = \vec{r} \times \vec{F}$
here:

\begin{aligned} \vec{τ} & = \vec{r} \times (+ q) \vec{E} + (- \vec{r}) \times (- q) \vec{E} \\ = 2 q \vec{r} \times \vec{E} \\ = q \vec{d} \times E \\ ∴ \vec{τ} & = \vec{p} \times \vec{E} \\ o r, | \vec{τ} | & = q d E \sin ϕ \end{aligned}

for linear motion: $W = \vec{F} \cdot \vec{d}$
for rotation: $d W = τ d ϕ = - p E \sin ϕ d ϕ$
rotate from $ϕ_{1} \to ϕ_{2}$ :

\begin{aligned} W & = - \int_{ϕ_{1}}^{ϕ_{2}} p E \sin ϕ . d ϕ \\ = + p E (\cos ϕ_{2} - \cos ϕ_{1}) \\ = - Δ (- p E \cos ϕ) \end{aligned}

$Δ E = Δ K + Δ U = 0 ⟹ W = - Δ U$
define potential energy:

\begin{aligned} U (ϕ) & = - p E \cos ϕ \\ U (ϕ) & = - \vec{p} \cdot \vec{E} \end{aligned}