New solution
en/10.1.25.md
+35 −0
| @@ -0,0 +1,35 @@ | |||
| + | ### Statement | ||
| + | |||
| + | $10.1.25.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
| + | |||
| + | To begin with, we will show the invariance of the sum in this case as well. The force acting on the electron inside the transition zone : | ||
| + | |||
| + | $$F = -B_r e \upsilon_z$$ | ||
| + | |||
| + | Where $B_r$ is the radial component of induction, $\upsilon_z$ is the component of velocity along the axis of symmetry. | ||
| + | |||
| + | $$\frac{dM}{dt} = -B_r e r \upsilon_z $$ | ||
| + | |||
| + | $$dM = - B_r e r dz$$ | ||
| + | |||
| + | Now let's consider the flow change through the cross section of a ring of arbitrary radius R: | ||
| + | |||
| + | $$d\Phi = 2 \pi R B_r dz$$ | ||
| + | |||
| + | From here: | ||
| + | |||
| + | $$dM = -\frac{1}{2\pi}ed\Phi$$ | ||
| + | |||
| + | $$M+\frac{1}{2\pi} e \Phi = const$$ | ||
| + | |||
| + | Well, now, it's very easy to find the desired moment, at first the moment of momentum is zero, because the electron does not rotate: | ||
| + | |||
| + | $$\frac{1}{2}eB_1 R^2 = M + \frac{1}{2}eB_2 R^2$$ | ||
| + | |||
| + | $$M = \frac{(B_2 - B_1)R^2e}{2}$$ | ||
| + | |||
| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
| @@ -0,0 +1,35 @@ | |||
| ### Statement | |||
| $10.1.25.$ [Insert the problem statement] | |||
| ### Solution | |||
| To begin with, we will show the invariance of the sum in this case as well. The force acting on the electron inside the transition zone : | |||
| $$F = -B_r e \upsilon_z$$ | |||
| Where $B_r$ is the radial component of induction, $\upsilon_z$ is the component of velocity along the axis of symmetry. | |||
| $$\frac{dM}{dt} = -B_r e r \upsilon_z $$ | |||
| $$dM = - B_r e r dz$$ | |||
| Now let's consider the flow change through the cross section of a ring of arbitrary radius R: | |||
| $$d\Phi = 2 \pi R B_r dz$$ | |||
| From here: | |||
| $$dM = -\frac{1}{2\pi}ed\Phi$$ | |||
| $$M+\frac{1}{2\pi} e \Phi = const$$ | |||
| Well, now, it's very easy to find the desired moment, at first the moment of momentum is zero, because the electron does not rotate: | |||
| $$\frac{1}{2}eB_1 R^2 = M + \frac{1}{2}eB_2 R^2$$ | |||
| $$M = \frac{(B_2 - B_1)R^2e}{2}$$ | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||