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Consider a conducting loop with resistance R and area A suspended by a non-conducting wire in a magnetic field [itex] \vec{B}=B\hat{y} [/itex]. The wire is a torsion spring with constant k.The equilibrium state of the loop is when it resides in the yz plane and its suspension is somehow that it can rotate around z axis with moment of inertia I.The loop is rotated by a very small angle [itex] \theta_0 [/itex] and is released.
This is how I tried to derive the equation of motion:
[itex]
i=-\frac{1}{R} \frac{d\phi_B}{dt}=-\frac{1}{R} \frac{d}{dt} BA\cos{\theta}=\frac{BA\dot{\theta}\sin{\theta}}{R} \Rightarrow i\approx \frac{BA\dot{\theta}\theta}{R}
[/itex]
[itex]
\vec{\tau}=i \vec{A}\times\vec{B}=\frac{BA\dot{\theta}\theta}{R} A\hat{x}\times B \hat{y} \Rightarrow \vec{\tau}=\frac{B^2A^2\dot{\theta}\theta}{R}\hat{z}
[/itex]
There is also the restoring torque of the torsion spring and so we have:
[itex]
\frac{B^2A^2\dot{\theta}\theta}{R}-k\theta=I\ddot{\theta} \Rightarrow IR\ddot{\theta}+(kR-B^2A^2\dot{\theta})\theta=0
[/itex]
The problem is,because [itex] \dot{\theta}_0=0 [/itex], the differential equation derived above isn't describing the motion of the loop at the first moment and only when the loop gains angular speed,the above DE can be used to describe its motion and so the answer to the above DE is contradictory when one applies the initial conditions to it.
What should I do?
Thanks
This is how I tried to derive the equation of motion:
[itex]
i=-\frac{1}{R} \frac{d\phi_B}{dt}=-\frac{1}{R} \frac{d}{dt} BA\cos{\theta}=\frac{BA\dot{\theta}\sin{\theta}}{R} \Rightarrow i\approx \frac{BA\dot{\theta}\theta}{R}
[/itex]
[itex]
\vec{\tau}=i \vec{A}\times\vec{B}=\frac{BA\dot{\theta}\theta}{R} A\hat{x}\times B \hat{y} \Rightarrow \vec{\tau}=\frac{B^2A^2\dot{\theta}\theta}{R}\hat{z}
[/itex]
There is also the restoring torque of the torsion spring and so we have:
[itex]
\frac{B^2A^2\dot{\theta}\theta}{R}-k\theta=I\ddot{\theta} \Rightarrow IR\ddot{\theta}+(kR-B^2A^2\dot{\theta})\theta=0
[/itex]
The problem is,because [itex] \dot{\theta}_0=0 [/itex], the differential equation derived above isn't describing the motion of the loop at the first moment and only when the loop gains angular speed,the above DE can be used to describe its motion and so the answer to the above DE is contradictory when one applies the initial conditions to it.
What should I do?
Thanks