Правка разделов «Statement», «Solution», «Answer»

naz правка от 26 июн. 2026 г., 11:24

правка #19260 предыдущая #19259 ← раньше

en/11.2.9.md +17 −4

		@@ -1,6 +1,9 @@
			### Statement

		−	$11.2.9.$ ~~[Insert the problem statement]~~
		+	$11.2.9.$ The figure shows flat shapes made of wire, the resistance of a unit of length of which is $1 Ohm/m$. Determine the currents in
		+	them if the shapes are placed in a uniform magnetic field that varies over time. The rate of change of the magnetic flux per unit area
		+	is $0.1 Wb/(m^2 ·s)$.
		+	![For problem $11.2.9$\|1844x666, 50%](../../img/11.2.9/Ð¡Ð½Ð¸Ð¼Ð¾Ðº ÑÐºÑÐ°Ð½Ð° 2026-06-26 142244.png)

			### Solution

			a) The area of the triangle:

			$$S = \frac{a^2 \sqrt{3}}{4}$$

			Denote by $v$ is the rate of change of flow through a unit area. Then the potential difference in the triangular frame is:

			$$U = \frac{a^2 \sqrt{3} v}{4}$$

			Frame resistance:

			$$R = 3\rho a$$

			Hence the current strength:

			$$I = \frac{U}{R} \approx 1.44 mA$$

			b) Considering our circle with a jumper as two circuits, the current in each circuit is the same and the current in the jumper goes towards each other, which means the current in the jumper is 0.

			Potential difference in the ring:

			$$U = \pi R^2 v$$

			Ring resistance:

			$$R = 2\pi R \rho$$

			The desired current in the arcs:

			$$I = \frac{U}{R} = 2.5 mA$$

			c) The potential difference to a greater extent and to a lesser extent, respectively:

			$$U_1 = (\frac{3}{4}\pi + \frac{1}{2})vR^2$$

			$$U_2 = (\frac{1}{4}\pi - \frac{1}{2})vR^2$$
		@@ -42,8 +45,8 @@Solution

			The resistances of the larger arc, the smaller arc, and the jumper, respectively:

		−	$$r_1 = \frac{3}{2}\piR \rho$$
		−	$$r_2 = \frac{1}{2}\piR \rho$$
		+	$$r_1 = \frac{3}{2}\pi R \rho$$
		+	$$r_2 = \frac{1}{2}\pi R \rho$$
			$$r_3 = \sqrt{2}R \rho$$

			So let's write down the Kirchhoff rule system.:

			$$I_1 - I_2 = I_3$$
			$$I_1 r_1 + I_3 r_3 = U_1$$
			$$I_2 r_2 - I_3 r_3 = U_2$$

			These equations are easier to solve by preparatory calculations of each voltage and resistance, I got the following currents:

			$$I_1 \approx 2.56 mA$$

			$$I_2 \approx 1.69 mA$$

			$$I_3 \approx 0.87 mA$$

			Savchenko's numbers are different, but very close to mine, if anyone has any ideas, write to me.
		@@ -64,4 +67,14 @@Solution

			#### Answer

		−	[Insert a concise answer or boxed result]
		+	a)
		+	$$I = \frac{U}{R} \approx 1.44 mA$$
		+	b)$$I = \frac{U}{R} = 2.5 mA$$
		+	Current in the jumper is 0.
		+
		+	c)
		+	$$I_1 \approx 2.56 mA$$
		+
		+	$$I_2 \approx 1.69 mA$$
		+
		+	$$I_3 \approx 0.87 mA$$

	@@ -1,6 +1,9 @@
	### Statement		### Statement

	$11.2.9.$ ~~[Insert the problem statement]~~		$11.2.9.$ The figure shows flat shapes made of wire, the resistance of a unit of length of which is $1 Ohm/m$. Determine the currents in
			them if the shapes are placed in a uniform magnetic field that varies over time. The rate of change of the magnetic flux per unit area
			is $0.1 Wb/(m^2 ·s)$.
			![For problem $11.2.9$\|1844x666, 50%](../../img/11.2.9/Ð¡Ð½Ð¸Ð¼Ð¾Ðº ÑÐºÑÐ°Ð½Ð° 2026-06-26 142244.png)

	### Solution		### Solution

	a) The area of the triangle:		a) The area of the triangle:

	$$S = \frac{a^2 \sqrt{3}}{4}$$		$$S = \frac{a^2 \sqrt{3}}{4}$$

	Denote by $v$ is the rate of change of flow through a unit area. Then the potential difference in the triangular frame is:		Denote by $v$ is the rate of change of flow through a unit area. Then the potential difference in the triangular frame is:

	$$U = \frac{a^2 \sqrt{3} v}{4}$$		$$U = \frac{a^2 \sqrt{3} v}{4}$$

	Frame resistance:		Frame resistance:

	$$R = 3\rho a$$		$$R = 3\rho a$$

	Hence the current strength:		Hence the current strength:

	$$I = \frac{U}{R} \approx 1.44 mA$$		$$I = \frac{U}{R} \approx 1.44 mA$$

	b) Considering our circle with a jumper as two circuits, the current in each circuit is the same and the current in the jumper goes towards each other, which means the current in the jumper is 0.		b) Considering our circle with a jumper as two circuits, the current in each circuit is the same and the current in the jumper goes towards each other, which means the current in the jumper is 0.

	Potential difference in the ring:		Potential difference in the ring:

	$$U = \pi R^2 v$$		$$U = \pi R^2 v$$

	Ring resistance:		Ring resistance:

	$$R = 2\pi R \rho$$		$$R = 2\pi R \rho$$

	The desired current in the arcs:		The desired current in the arcs:

	$$I = \frac{U}{R} = 2.5 mA$$		$$I = \frac{U}{R} = 2.5 mA$$

	c) The potential difference to a greater extent and to a lesser extent, respectively:		c) The potential difference to a greater extent and to a lesser extent, respectively:

	$$U_1 = (\frac{3}{4}\pi + \frac{1}{2})vR^2$$		$$U_1 = (\frac{3}{4}\pi + \frac{1}{2})vR^2$$

	$$U_2 = (\frac{1}{4}\pi - \frac{1}{2})vR^2$$		$$U_2 = (\frac{1}{4}\pi - \frac{1}{2})vR^2$$
	@@ -42,8 +45,8 @@Solution

	The resistances of the larger arc, the smaller arc, and the jumper, respectively:		The resistances of the larger arc, the smaller arc, and the jumper, respectively:

	$$r_1 = \frac{3}{2}\piR \rho$$		$$r_1 = \frac{3}{2}\pi R \rho$$
	$$r_2 = \frac{1}{2}\piR \rho$$		$$r_2 = \frac{1}{2}\pi R \rho$$
	$$r_3 = \sqrt{2}R \rho$$		$$r_3 = \sqrt{2}R \rho$$

	So let's write down the Kirchhoff rule system.:		So let's write down the Kirchhoff rule system.:

	$$I_1 - I_2 = I_3$$		$$I_1 - I_2 = I_3$$
	$$I_1 r_1 + I_3 r_3 = U_1$$		$$I_1 r_1 + I_3 r_3 = U_1$$
	$$I_2 r_2 - I_3 r_3 = U_2$$		$$I_2 r_2 - I_3 r_3 = U_2$$

	These equations are easier to solve by preparatory calculations of each voltage and resistance, I got the following currents:		These equations are easier to solve by preparatory calculations of each voltage and resistance, I got the following currents:

	$$I_1 \approx 2.56 mA$$		$$I_1 \approx 2.56 mA$$

	$$I_2 \approx 1.69 mA$$		$$I_2 \approx 1.69 mA$$

	$$I_3 \approx 0.87 mA$$		$$I_3 \approx 0.87 mA$$

	Savchenko's numbers are different, but very close to mine, if anyone has any ideas, write to me.		Savchenko's numbers are different, but very close to mine, if anyone has any ideas, write to me.
	@@ -64,4 +67,14 @@Solution

	#### Answer		#### Answer

	[Insert a concise answer or boxed result]		a)
			$$I = \frac{U}{R} \approx 1.44 mA$$
			b)$$I = \frac{U}{R} = 2.5 mA$$
			Current in the jumper is 0.

			c)
			$$I_1 \approx 2.56 mA$$

			$$I_2 \approx 1.69 mA$$

			$$I_3 \approx 0.87 mA$$