Dipole Moment

Topics

• Lamor Frequency
• Polarization

$\vec{\mu }=q\vec{d}$

• $\mu$ is the dipole moment in debye
• $q$ is the magnitude of the separated charge in Coloumbs
• $\vec{d}$ is a vector pointing from $q_{-}$ to $q_{}$

Electromagnetism

$\vec{p}=\sum _{i=1}^n{q}_i{\vec{r}}_i^{\prime }$

• Dipole moment of a set of charges
• ${\vec{r}}^{\prime }$ is the source vector
• Depends on how origin is defined for physical dipole as opposed to ideal dipole

$\vec{p}\equiv \int {\vec{r}}^{\prime }\rho ({\vec{r}}^{\prime })\mathrm{d}{\tau }^{\prime }$

• Dipole moment of a distribution
• ${\tau }^{\prime }$ is a volume element

${\vec{E}}_{\text{dip}}(r,\theta )=\frac{kp}{r^3}(2\cos \theta \hat{r}+\sin \theta \hat{\theta })$

• Electric field produced by an dipole moment
• $p$ is the magnitude of the dipole moment
• $r$ is the distance from the dipole moment’s center to the point of measurement
• $\theta$ is the angle between the dipole vector and the point of measurement (so imagine pointing the dipole moment in the same direction as the z-axis)

$\vec{\tau }=\vec{p}\times \vec{E}$

• Torque $\vec{\tau }$ experienced by a dipole moment $\vec{p}$ in an electric field $\vec{E}$

$\vec{F}=(\vec{p}\cdot \nabla )\vec{E}$

• The force

Quantum Physics

$\vec{\mu }=\gamma \vec{S}$

• $\gamma$ is the gyromagnetic ratio
• $S$ is the spin angular momentum

$\hat{H}=-\vec{\mu }\cdot \vec{B}=-\gamma \vec{B}\cdot \vec{S}$

$\vec{\tau }=\vec{\mu }\times \vec{B}$

$\vec{F}=\nabla (\vec{\mu }\cdot \vec{B})$

• Force due to an inhomogeneous magnetic field on a dipole $\vec{\mu }$