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| ### Statement |
| ### Statement |
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| $11.5.23.$ A copper ring of radius $r$ and mass $m$ hangs on a thread, making small torsional vibrations with a period of $T$. The inductance of the ring is $L$. How will the oscillation period of the ring change if it is placed in a horizontal homogeneous magnetic field of induction $B$ parallel to the plane of the ring in the equilibrium position? The moment of inertia of the ring relative to the axis passing through the diameter is equal to $J$. Ignore the resistance of the ring. |
| $11.5.23.$ A copper ring of radius $r$ and mass $m$ hangs on a thread, making small torsional vibrations with a period of $T$. The inductance of the ring is $L$. How will the oscillation period of the ring change if it is placed in a horizontal homogeneous magnetic field of induction $B$ parallel to the plane of the ring in the equilibrium position? The moment of inertia of the ring relative to the axis passing through the diameter is equal to $J$. Ignore the resistance of the ring. |
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| ### Solution |
| ### Solution |
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| The oscillation period of a conventional torsional pendulum with a small amplitude is: |
| The oscillation period of a conventional torsional pendulum with a small amplitude is: |
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| $$T = 2\pi \sqrt{\frac{J}{k}}$$ | | $$T = 2\pi \sqrt{\frac{J}{k}}$$ |
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| Where $k$ is the torsional stiffness of the thread, we express it: | | Where $k$ is the torsional stiffness of the thread, we express it: |
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| $$k = \frac{4 \pi^2 J}{T^2}$$ | | $$k = \frac{4 \pi^2 J}{T^2}$$ |
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| The moment acting on the frame from the side of the magnetic field is equal to: | | The moment acting on the frame from the side of the magnetic field is equal to: |
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| $$M = \mu \times B \approx \mu B$$ | | $$M = \mu \times B \approx \mu B$$ |
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| where $\mu = \pi r^2I$ is the magnetic moment of the frame. | | where $\mu = \pi r^2I$ is the magnetic moment of the frame. |
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| The magnetic flux through the frame is: | | The magnetic flux through the frame is: |
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| $$\Phi = L I = B \pi r^2 \varphi$$ | | $$\Phi = L I = B \pi r^2 \varphi$$ |
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| $$I = \frac{B \pi r^2}{L} \varphi$$ | | $$I = \frac{B \pi r^2}{L} \varphi$$ |
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| Let's make up the equation of oscillations in 2ZN for rotational motion: | | Let's make up the equation of oscillations in 2ZN for rotational motion: |
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| $$J \beta = k = \frac{4 \pi^2 J}{T^2} \varphi + \frac{B^2 \pi^2 r^4}{L} \varphi$$ | | $$J \beta = k = \frac{4 \pi^2 J}{T^2} \varphi + \frac{B^2 \pi^2 r^4}{L} \varphi$$ |
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| The period is similar to the usual fluctuations, after the transformations: | | The period is similar to the usual fluctuations, after the transformations: |
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| $$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$ | | $$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$ |
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| #### Answer | | #### Answer |
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| $$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$ | | $$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$ |
