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## Homework Statement

A circular disk rotates about its axis with angular velocity ##\omega##. The disk is made of metal with

conductivity g, and its thickness is t. The rotating disk is placed between the pole' faces of a magnet

which produces a uniform magnetic field B over a small square area of size a^2 at the average distance r from the axis; B is perpendicular to the disk. Calculate the approximate torque on the disk.

(Make a reasonable assumption about the resistance of the "eddy current circuit. ").

## Homework Equations

Force density=##\vec{J}\wedge\vec{B}##

Current density is given by Ohm's law: ##\vec{J}=g\vec{E}##, where g is the conductivity

In our case ##\vec{J}=g\vec{V}\wedge\vec{B}=gVB\hat{a}_r##, where ##\hat{a}_r## is the radial unit vector of a cilyndrical coordenate system and ##\vec{V}\wedge\vec{B}## is equivalent electri field

## The Attempt at a Solution

The total force is

$$\int\vec{J}\wedge\vec{B}\,dr^3=\int gVB\,\hat{a}_r\wedge B\,\hat{k}\,dr^3=\int gVB^2(-\hat{a}_\theta)\,dr^3\approx -gVB^2\hat{a}_\theta\int dr^3=-gVB^2 a^2 t \hat{a}_\theta $$

putting ##V=\omega r## the torque of this force is:

$$-g\omega r^2 B^2 a^2 t \,\hat{a}_\theta $$This is a problem from Electromagnetic Theory of Reitz-Milford-Christy and the answer given in the book is

$$-\frac{1}{2}g\omega r^2 B^2 a^2 t \,\hat{a}_\theta $$

I don't know where the factor ##\frac{1}{2}## comes from