Edits to “Statement”, “Answer”

naz edited Jun 15, 2026, 8:25 PM

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revision #18863 parent #18862 ← older

en/10.1.28.md +2 −3

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

		−	$10.1.28.$ [Insert the problem statement]
		+	$10.1.28.$ Determine the minimum radius that an electron beam can have when moving from the field with induction $B_1$ to the field with induction $B_2$. The axes of symmetry of the transition field and the beam coincide. The beam radius in the B1 field is $R$, and the beam velocity in the $B_1$ field is parallel to the induction.

			### Solution

			We will use the same invariant.

			At the beginning, the angular momentum is zero.

			Now we are talking about the minimum width condition, obviously, since this is the minimum, the radial component of the velocity is 0. But also the minimum condition is that the longitudinal velocity is 0, because we are considering the boundary case when the beam remains focused after the transition, which means:

			$$\frac{eB_1R^2}{2} = \frac{eB_2r^2}{2}$$

			$$r = R\sqrt{\frac{B_1}{B_2}}$$

		@@ -17,5 +17,4 @@Solution
			Savchenko, apparently, has a typo.

			#### Answer
		−
		−	[Insert a concise answer or boxed result]
		+	$$r = R\sqrt{\frac{B_1}{B_2}}$$

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

	$10.1.28.$ [Insert the problem statement]		$10.1.28.$ Determine the minimum radius that an electron beam can have when moving from the field with induction $B_1$ to the field with induction $B_2$. The axes of symmetry of the transition field and the beam coincide. The beam radius in the B1 field is $R$, and the beam velocity in the $B_1$ field is parallel to the induction.

	### Solution		### Solution

	We will use the same invariant.		We will use the same invariant.

	At the beginning, the angular momentum is zero.		At the beginning, the angular momentum is zero.

	Now we are talking about the minimum width condition, obviously, since this is the minimum, the radial component of the velocity is 0. But also the minimum condition is that the longitudinal velocity is 0, because we are considering the boundary case when the beam remains focused after the transition, which means:		Now we are talking about the minimum width condition, obviously, since this is the minimum, the radial component of the velocity is 0. But also the minimum condition is that the longitudinal velocity is 0, because we are considering the boundary case when the beam remains focused after the transition, which means:

	$$\frac{eB_1R^2}{2} = \frac{eB_2r^2}{2}$$		$$\frac{eB_1R^2}{2} = \frac{eB_2r^2}{2}$$

	$$r = R\sqrt{\frac{B_1}{B_2}}$$		$$r = R\sqrt{\frac{B_1}{B_2}}$$

	@@ -17,5 +17,4 @@Solution
	Savchenko, apparently, has a typo.		Savchenko, apparently, has a typo.

	#### Answer		#### Answer
			$$r = R\sqrt{\frac{B_1}{B_2}}$$
	[Insert a concise answer or boxed result]