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.