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

Homework

1. 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.

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

(Hint: For a current carrying wire, $d\vec{F=Id\vec{\ell} \times \vec{B}$})

2. 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?

3. Find the torque on each side of the wire loop due to the magnetic field.

4. Find the net torque on the wire loop. Consider the Physical Implication: What does this result mean for the motion of the loop?

5. 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.)

6. 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?

2. \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.

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