- #1

Happiness

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## Homework Statement

Problem 6.2 of Griffith's "Introduction to Electrodynamics": Starting from the Lorentz force law ##\vec F=\int I (d\vec l \times \vec B)##, show that the torque on any steady current distribution (not just a square loop) in a uniform field ##\vec B## is ##\vec m\times \vec B##. (##\vec m## is the magnetic moment.)

## Homework Equations

Let the torque be ##\vec N##.

##d\vec N = \vec r\times d\vec F##.

## The Attempt at a Solution

Useful identity: ##\oint \vec r\times d\vec l = 2\vec a##, where ##\vec a## is the area of the loop and points perpendicularly to its surface.

My question: the solution says that ##d\vec r = d\vec l##, which I don't understand. They are clearly pointing in different directions. ##d\vec r## points in the direction from the origin to the point ##r##, while ##d\vec l## points in the direction of the wire of the loop, which in general is different from the direction of ##d\vec r##.

The solution: