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1. 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 a^{2}. Find the approximate torque tending to slow the disk at the instant its angular velocity is ω.
2. Homework Equations
3. The Attempt at a Solution
[itex]\xi = B \frac{dA}{dt} [/itex]
[itex] = B a \frac{da}{dt} [/itex]
(Right here ^ Can I do this?)
[itex] = B a v [/itex]
[itex] = B a \omega dr [/itex]
[itex] \Rightarrow \int^{r+a/2}_{ra/2} B a \omega dr [/itex]
[itex] =  B a^2 \omega [/itex]
And
[itex] R = \frac{L}{\sigma A} = \frac{a}{\sigma ab} = \frac{1}{\sigma b}[/itex]


Thus
[itex]F = i a \times B[/itex]
[itex] = \frac{\xi}{R} a \times B[/itex]
[itex] F =  B^2 a^2 \omega \sigma b [/itex]


[itex]\tau = \int F dr[/itex]
[itex]\tau =  B^2 a^2 \omega \sigma b \int^{r+a/2}_{ra/2} dr[/itex]
[itex]\tau =  B^2 a^3 \omega \sigma b[/itex]
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 a^{2}. Find the approximate torque tending to slow the disk at the instant its angular velocity is ω.
2. Homework Equations
3. The Attempt at a Solution
[itex]\xi = B \frac{dA}{dt} [/itex]
[itex] = B a \frac{da}{dt} [/itex]
(Right here ^ Can I do this?)
[itex] = B a v [/itex]
[itex] = B a \omega dr [/itex]
[itex] \Rightarrow \int^{r+a/2}_{ra/2} B a \omega dr [/itex]
[itex] =  B a^2 \omega [/itex]
And
[itex] R = \frac{L}{\sigma A} = \frac{a}{\sigma ab} = \frac{1}{\sigma b}[/itex]


Thus
[itex]F = i a \times B[/itex]
[itex] = \frac{\xi}{R} a \times B[/itex]
[itex] F =  B^2 a^2 \omega \sigma b [/itex]


[itex]\tau = \int F dr[/itex]
[itex]\tau =  B^2 a^2 \omega \sigma b \int^{r+a/2}_{ra/2} dr[/itex]
[itex]\tau =  B^2 a^3 \omega \sigma b[/itex]
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