Calculate Torque of Eddy Current Brake for Rotating Disk with Magnetic Field

[AJKing]

Homework Statement

Refer to Figure attached.

A disk of conductivity σ and thickness b rotates around an axis through its center with a magnetic field B applied perpendicular to the plane of the disk over a small area a². Find the approximate torque tending to slow the disk at the instant its angular velocity is ω.

Homework Equations

The Attempt at a Solution

$\xi = -B \frac{dA}{dt}$

$= -B a \frac{da}{dt}$

(Right here ^ Can I do this?)

$= -B a v$

$= -B a \omega dr$

$\Rightarrow \int^{r+a/2}_{r-a/2} -B a \omega dr$

$= - B a^2 \omega$

And

$R = \frac{L}{\sigma A} = \frac{a}{\sigma ab} = \frac{1}{\sigma b}$

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Thus

$F = i a \times B$

$= \frac{\xi}{R} a \times B$

$F = - B^2 a^2 \omega \sigma b$

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$\tau = \int F dr$

$\tau = - B^2 a^2 \omega \sigma b \int^{r+a/2}_{r-a/2} dr$

$\tau = - B^2 a^3 \omega \sigma b$

 

[AJKing]

I think I've handled my integrals poorly...
will revise.