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## Lorentz Force and Work Done on a Rectangular Loop: Instructor's Guide

### Main Ideas

- In a homogeneous magnetic field, the net force on a current-carrying loop is zero (although torque is non-zero, which leads to precession)
- The work done to rotate a loop can be written as a dot product between the magnetic moment and the magnetic field ($U_m=\vec{\mu}\cdot\vec{B}$ for $U_m=0$ corresponding to the magnetic moment aligned with the magnetic field).
- This calculation is covered well in Matter and Interactions § 21.7-21.8

### Students' Task

Students are placed into small groups and asked to calculate the force on a rectangular loop with a current moving through it due to an external magnetic field. After finding a general expression for the force, the groups must then calculate the amount of energy needed to rotate the loop through some angle.

### Prerequisite Knowledge

- Lorentz Force Law on a current-carrying wire
- Relationship between force and work

### Props/Equipment

- Tabletop Whiteboard with markers
- Rectangular Loop

### Activity: Introduction

After reminding students of the Lorentz Force Law (either through a mini-lecture or with a SWBQ), they are given a handout that asks them to consider the force on a rectangular loop of current-carrying wire due to an external magnetic field and then the work needed to rotate the wire through some angle.

### Activity: Student Conversations

**Lorentz Force Law in terms of current**:- If you motivate this activity with the Stern-Gerlach, students sometimes fail to make the connection between electrons going around a ring and the conventional current, which is the movement of positive charges (they add in an extra minus sign).
- In the earlier paradigms, students spend a lot of time talking about current in terms of charge distribution and so they should be able to go from $\vec{I}=\lambda\vec{v}$ and $\lambda=\frac{q}{L}$ to $q\vec{v}=\vec{I}L$ to $\vec{F}=q\vec{v}\times\vec{B}=\vec{I}L\times\vec{B}$. However, students struggle in both remembering the relevant relationships and performing this chain of reasoning.
- To speed up the activity, one can incorporate this into the initial mini-lecture or SWBQ on Lorentz Force Law.

**Net force v. net torque**: some students are confused and troubled by the fact that the net force is zero, but the net torque is non-zero. It can help to remind them that net force results in a change in linear momentum, whereas net torque produces a change in angular momentum.**Relationship between force and work**: $W=-\int \vec{F}\cdot\vec{dr}$- In trying to remember the relationship, students tend to fall back on $W=Fd$ which they remember from their intro classes. It is important to remind them that it is the part of the force that is parallel to the motion. This is addressed naturally if you use $\vec{dr}$ to do the line integral (see the Vector Differential activity and Geometry of Vector Calculus § The Vector Differential, § Coordinate Expressions, and § Scalar Line Integrals)
- Some students do struggle to remember how to take the dot product between something in the $\hat{x}$ direction ($\vec{F}$) and something in the $\hat{\theta}$ direction ($\vec{dr}$).
- In order for the signs to work out, emphasize that it is the work that “you” do to rotate the loop

### Activity: Wrap-up

After establishing that the work done is $W=-IwlB\cos\theta$, introduce the idea of the magnetic dipole moment, $\vec{\mu}=IA$ in the direction perpendicular to the plane of the loop in order to get to the potential energy for a magnetic dipole: $U_m=-\vec{\mu}\cdot\vec{B}$. This then leads naturally into the discussion of what happens when the magnetic field is inhomogeneous, i.e. the Stern-Gerlach experiment, where the net force is no longer zero and the loop will move up or down depending on its orientation.

### Extensions

This is a part of a sequence of activities designed to introduce the Stern-Gerlach experiment.