# Magnetic Momentum

some familiarity with force, torque, magnetic fields, vector dot product, vector cross product, gradient, current

## In-class Content

### Lecture: Magnetic Moment

- Magnetic Moment: conceptual definition - tells you about torque response in an external magnetic field ($\vec{\tau}=\vec{\mu}\times\vec{B}$)
- formal bits: symbol $\vec{\mu}$, vector, units/dimensions (charge*length^2; torque/magnetic field; energy/magnetic field)
- magnetic moment for a current loop: $\vec{\mu}=I\vec{A}$
- Small Calculation: A particle with mass $m$ and charge $q$ moves at speed $v$ in a circle of radius $R$. What is the magnetic moment of the particle? What is the angular momentum of the particle? What is $\vec{\mu}$ in terms of $\vec{L}$?
- force and torque on a current loop in a magnetic field
- magnetic momentum and angular momentum for a charged spinning sphere

### SWBQ Sequence: Spinning Charged Sphere in a Magnetic Field

### Lecture: Stern Gerlach Experiment

- Stern Gerlach Experiment: history and classical prediction
- Let students play with Stern-Gerlach PhET
- Introduce OSP Stern-Gerlach Simulation

### Homework

Consider a square wire loop with sides of length $L$ carrying current $I$. The normal to the plane of the wire loop is at an angle $\theta$ with respect to a uniform magnetic field $\vec B$. Take the direction of the magnetic field to be $\hat{z}$, the origin of coordinates to be at the center of the loop, the high side of the wire to be at constant positive values of $x$, and the current to be flowing counter-clockwise if looking down along the $z$-axis.

Find the force on each side the wire loop due to the magnetic field.

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*Hint: For a current carrying wire, $d\vec{F*=Id\vec{\ell} \times \vec{B}$})Find the net force on the loop.

*Consider the Physical Implication:*What does this result mean for the motion of the loop?*Compare \& Contrast Systems:*How does this result compare/contrast with the example we did in class?Find the torque on each side of the wire loop due to the magnetic field.

Find the net torque on the wire loop.

*Consider the Physical Implication:*What does this result mean for the motion of the loop?Show that the (potential) energy $H$ of the wire loop in the external magnetic field is given by: $$H=-\vec{\mu}\cdot\vec{B}$$

(

*Hint: To find the work done by a torque during a rotation, integrate the torque over the rotation angle.*)*Examine Special Cases:*For what configuration of the loop and field would you expect the energy to be minimum? Maximum? Does the energy equation agree with your analysis?

\begin{enumerate}

\item Explain the key features of the Stern-Gerlach experiment. (What features make the experiment measure what it is supposed to measure?) \item \textit{Contrast Classical/Quantum} Explain what you would predict based only on classical physics for the Stern-Gerlach experiment and describe the difference between the classical prediction and the actual experimental results.