Translated 9.1.6-9.1.15

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+ <meta name="description" content="a) A wire frame in the form of a circle with current can rotate about the horizontal axis OO^\prime. The mass of a unit length of the wire is \rho, the current in the frame is I. The frame is in a magnetic field of induction B, directed along the gravitational field. Determine the angle of deviation of the plane of the circle from the vertical. b) A wire frame in the form of a circle has a wire jumper along the diameter, parallel to the horizontal axis OO^\prime, about which the frame can rotate. The mass of a unit length of the frame and the jumper is the same and is equal to \rho. The current entering the frame is equal to I. The frame is in a magnetic field of induction B, directed parallel to the gravitational field. At what angle from the vertical will the frame deviate?">
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+ <meta property="og:title" content="a) A wire frame in the form of a circle with current can rotate about the horizontal axis OO^\prime. The mass of a unit length of the wire is \rho, the current in the frame is I. The frame is in a magnetic field of induction B, directed along the gravitational field. Determine the angle of deviation of the plane of the circle from the vertical. b) A wire frame in the form of a circle has a wire jumper along the diameter, parallel to the horizontal axis OO^\prime, about which the frame can rotate. The mass of a unit length of the frame and the jumper is the same and is equal to \rho. The current entering the frame is equal to I. The frame is in a magnetic field of induction B, directed parallel to the gravitational field. At what angle from the vertical will the frame deviate?">
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+ <meta property="og:description" content="a) A wire frame in the form of a circle with current can rotate about the horizontal axis OO^\prime. The mass of a unit length of the wire is \rho, the current in the frame is I. The frame is in a magnetic field of induction B, directed along the gravitational field. Determine the angle of deviation of the plane of the circle from the vertical. b) A wire frame in the form of a circle has a wire jumper along the diameter, parallel to the horizontal axis OO^\prime, about which the frame can rotate. The mass of a unit length of the frame and the jumper is the same and is equal to \rho. The current entering the frame is equal to I. The frame is in a magnetic field of induction B, directed parallel to the gravitational field. At what angle from the vertical will the frame deviate?">
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+ <title>a) A wire frame in the form of a circle with current can rotate about the horizontal axis OO^\prime. The mass of a unit length of the wire is \rho, the current in the frame is I. The frame is in a magnetic field of induction B, directed along the gravitational field. Determine the angle of deviation of the plane of the circle from the vertical. b) A wire frame in the form of a circle has a wire jumper along the diameter, parallel to the horizontal axis OO^\prime, about which the frame can rotate. The mass of a unit length of the frame and the jumper is the same and is equal to \rho. The current entering the frame is equal to I. The frame is in a magnetic field of induction B, directed parallel to the gravitational field. At what angle from the vertical will the frame deviate?</title>
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+ <h3 id="back-link"><a href="../../#9.1">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $9.1.11.$ a) A wire frame in the form of a circle with current can rotate about the horizontal axis $OO^\prime$. The mass of a unit length of the wire is $\rho$, the current in the frame is $I$. The frame is in a magnetic field of induction B, directed along the gravitational field. Determine the angle of deviation of the plane of the circle from the vertical. </p><p>
+b) A wire frame in the form of a circle has a wire jumper along the diameter, parallel to the horizontal axis $OO^\prime$, about which the frame can rotate. The mass of a unit length of the frame and the jumper is the same and is equal to $\rho$. The current entering the frame is equal to $I$. The frame is in a magnetic field of induction $B$, directed parallel to the gravitational field. At what angle from the vertical will the frame deviate?
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+<center>
+ <figure>
+ <img src="https://savchenkosolutions.com/9/9.1.11/statement.png"
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+ <figcaption>
+ For problem $9.1.11$
+ </figcaption>
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+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ a) Using the expression for the moment of magnetic field obtained in <a href="../9.1.10">9.1.10</a>, $\vec{N} = [\vec{M} \times \vec{B}]$ we obtain
+$$ N = BIS\cdot \cos\alpha = \pi R^2BI \cdot \cos\alpha $$
+in this case the moment of gravity:
+$$ M_{mg} = Rmg\cdot \cos\alpha = 2\pi R^2\rho g\cdot \sin\alpha $$
+these moments are equal:
+$$ 2\pi R^2\rho g \cdot \sin\alpha = \pi BI R^2 \cdot \cos\alpha $$
+after transformations, we obtain
+$$ \boxed{\tan\alpha = \frac{BI}{2\rho g}} $$
+b) Find the moment of gravity:
+$$ M_{mg} = mgR\cdot \sin\alpha = (2\pi R + 2R)\rho gR \cdot \sin\alpha $$
+
+$$ M_{mg} = 2R^2\rho g(1+\pi ) \cdot \sin\alpha $$
+Now, we represent the initial circuit as two superimposed circuits with currents $I_1$ and $I_2$ such that $I = I_1 + I_2$, the circuit with $I_2$ is a semicircle with a partition, the circuit with $I_1$ is a ring without a partition, then from the parallel connection it follows that: $I_1\cdot \pi R = I_2 \cdot 2R$ then
+$$ I_1 = I_2\cdot \frac{2}{\pi} $$
+
+$$ I = I_1 + I_2 = I_2\left(1+\frac{2}{\pi}\right) $$
+
+$$ I_1 = I\cdot \frac{2}{\pi +2}; \quad $$
+Then the moments for these contours are:
+$$ M_{k1} = BS_1\cdot \cos\alpha =\pi R^2I\cdot \frac{2}{\pi +2}B\cdot \cos\alpha $$
+
+$$ M_{k2} = BS_2\cdot \cos\alpha = \frac{1}{2}\pi R^2I\cdot \frac{\pi}{\pi +2}B\cdot \cos\alpha $$
+Total moment of the contours:
+$$ M_k = \frac{1}{2}\pi R^2 BI \cdot \cos\alpha\cdot \frac{\pi + 4}{\pi +2} $$
+Let's equate this moment and the moment of gravity:
+$$ \frac{1}{2}\pi R^2 BI \cdot \cos\alpha\cdot \frac{\pi + 4}{\pi +2} = 2R^2\rho g(1+\pi ) \cdot \sin\alpha $$
+from here
+$$ \boxed{\tan\alpha = \frac{\pi (\pi +4)}{4(\pi +2)(\pi +1)} \cdot \frac{BI}{\rho g}} $$
+
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$\tan\alpha = \frac{\pi (\pi +4)}{4(\pi +2)(\pi +1)} \cdot \frac{BI}{\rho g}$$
+ </p>
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+ Andrei Yersh<br>
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