Ethan's Notes

    Dipole Moment

    #Physics #Chemistry

    Topics

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

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

    Electromagnetism

    $\displaystyle \vec{p}=\sum_{i = 1}^{n}q_{i}\vec{r}_{i}'$

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

    $\displaystyle \vec{p}\equiv \int \vec{r}' \rho(\vec{r}'), \mathrm{d}\tau'$

    • Dipole moment of a distribution
    • $\displaystyle \tau'$ is a volume element

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

    • Electric field produced by an dipole moment
    • $\displaystyle p$ is the magnitude of the dipole moment
    • $\displaystyle r$ is the distance from the dipole moment's center to the point of measurement
    • $\displaystyle \theta$ 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)

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

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

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

    • The force

    Quantum Physics

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

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

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

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

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