A square frame with current is located in a uniform magnetic field of induction B. The frame mass is m, and the current in it is I. Determine the frequency of free oscillations of the frame around the OO^\prime axis.

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    <h3 id="back-link"><a href="/#9.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $9.1.8^*.$ A square frame with current is located in a uniform magnetic field of induction $B$. The frame mass is $m$, and the current in it is $I$. Determine the frequency of free oscillations of the frame around the $OO^\prime$ axis.

    <h3>Solution</h3>
    <p>
        Let's consider a small deviation from equilibrium by an angle $\alpha$: </p><p>

The moment acting on the frame from the magnetic field:
$$M = BIS\cdot \sin\alpha = BIa^2 \sin\alpha$$
where $a$ is the side of the frame

gravity does not create a moment as it passes through the point of rotation

From the basic equation of rotational motion:
$$J\ddot{\alpha} = \sum\pm{M_i}$$
for our case:
$$J\ddot{\alpha} = -BIa^2 \sin\alpha\quad (1)$$
now we will find the moment of inertia $J$ of the frame during the observed oscillations

The moment of inertia of the upper and lower sides of the frame is
$$J_1=J_2 = \frac{m}{4}\cdot \left(\frac{a}{2}\right)^2 = \frac{m a^2}{16}\quad (2)$$
where $a/2$ is the distance to this side.

Moment of inertia of the lateral sides:
$$J_3 = J_4 = \frac{1}{12} \cdot \frac{m}{4}\cdot a^2 = \frac{m a^2}{48}\quad (3)$$
We find the moment of inertia $J$ by substituting $(2)$ and $(3)$
$$J = J_1 + J_2 + J_3 + J_4$$

$$J = 2\cdot \frac{m a^2}{16} + 2 \cdot \frac{m a^2}{48} = \frac{m a^2}{6}$$
then equation $(1)$ will look like:
$$\frac{m a^2}{6}\ddot{\alpha} = -BIa^2 \sin\alpha$$

$$\ddot{\alpha} = -\frac{6BI}{m} \alpha (t)$$
Solving the equation of harmonic oscillations, we find the oscillation frequency
$$\boxed{\omega = \sqrt{\frac{6BI}{m}}}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$\omega = \sqrt{\frac{6BI}{m}}$$
    </p>
	<p style="text-align: right; font-style: italic; font-size: 14;">
      Andrei Yersh<br>
    </p>


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