Edits to “Statement”, “Solution”, “Answer”
en/11.2.16.md
+4 −4
| @@ -1,10 +1,10 @@ | |||
| ### Statement | |||
| − | $11.2.16.$ [Insert the problem statement] | ||
| + | $11.2.16.$ In a uniform magnetic field, an electron moves around a circle of a certain radius. Does the radius of curvature of the electron's trajectory decrease or increase with a slow increase in the magnetic field induction? | ||
| ### Solution | |||
| − |  | ||
| The magnetic force $evB$ provides centripetal acceleration to the electron, so | |||
| \[evB=\frac{mv^2}{r}\qquad\Rightarrow\qquad r=\frac{mv}{eB},\] | |||
| where $-e$ and $m$ are the charge and the mass of the electron, respectively. Thus in order for the electron to maintain the same circular orbit, we must have | |||
| \[\frac{v+\Delta v}{B+\Delta B}=\frac{v}{B}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{v}{B}=\frac{er}{m}.\] | |||
| The change in electron's speed is caused by induced electric field | |||
| \[E=\frac{1}{2\pi r}\cdot\pi r^2\cdot\frac{\Delta B}{\Delta t}=\frac{r}{2}\cdot\frac{\Delta B}{\Delta t}\] | |||
| as a result of changing magnetic flux. Thus, | |||
| @@ -22,8 +22,8 @@Solution | |||
| \[\Delta v=\frac{eE\Delta t}{m}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{er}{2m}.\] | |||
| − | (Note that the electric force on the electron is in the opposite direction to the electric field.) Therefore, the additional speed caused by induced electric field is not sufficient to keep the electron in the same orbit, and the radius of curvature of | ||
| + | (Note that the electric force on the electron is in the opposite direction to the electric field.) Therefore, the additional speed caused by induced electric field is not sufficient to keep the electron in the same orbit, and the radius of curvature of its trajectory will decrease. | ||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | Decrease | ||
| @@ -1,10 +1,10 @@ | |||
| ### Statement | ### Statement | ||
| $11.2.16.$ [Insert the problem statement] | $11.2.16.$ In a uniform magnetic field, an electron moves around a circle of a certain radius. Does the radius of curvature of the electron's trajectory decrease or increase with a slow increase in the magnetic field induction? | ||
| ### Solution | ### Solution | ||
|  | ||
| The magnetic force $evB$ provides centripetal acceleration to the electron, so | The magnetic force $evB$ provides centripetal acceleration to the electron, so | ||
| \[evB=\frac{mv^2}{r}\qquad\Rightarrow\qquad r=\frac{mv}{eB},\] | \[evB=\frac{mv^2}{r}\qquad\Rightarrow\qquad r=\frac{mv}{eB},\] | ||
| where $-e$ and $m$ are the charge and the mass of the electron, respectively. Thus in order for the electron to maintain the same circular orbit, we must have | where $-e$ and $m$ are the charge and the mass of the electron, respectively. Thus in order for the electron to maintain the same circular orbit, we must have | ||
| \[\frac{v+\Delta v}{B+\Delta B}=\frac{v}{B}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{v}{B}=\frac{er}{m}.\] | \[\frac{v+\Delta v}{B+\Delta B}=\frac{v}{B}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{v}{B}=\frac{er}{m}.\] | ||
| The change in electron's speed is caused by induced electric field | The change in electron's speed is caused by induced electric field | ||
| \[E=\frac{1}{2\pi r}\cdot\pi r^2\cdot\frac{\Delta B}{\Delta t}=\frac{r}{2}\cdot\frac{\Delta B}{\Delta t}\] | \[E=\frac{1}{2\pi r}\cdot\pi r^2\cdot\frac{\Delta B}{\Delta t}=\frac{r}{2}\cdot\frac{\Delta B}{\Delta t}\] | ||
| as a result of changing magnetic flux. Thus, | as a result of changing magnetic flux. Thus, | ||
| @@ -22,8 +22,8 @@Solution | |||
| \[\Delta v=\frac{eE\Delta t}{m}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{er}{2m}.\] | \[\Delta v=\frac{eE\Delta t}{m}\qquad\Rightarrow\qquad\frac{\Delta v}{\Delta B}=\frac{er}{2m}.\] | ||
| (Note that the electric force on the electron is in the opposite direction to the electric field.) Therefore, the additional speed caused by induced electric field is not sufficient to keep the electron in the same orbit, and the radius of curvature of |
(Note that the electric force on the electron is in the opposite direction to the electric field.) Therefore, the additional speed caused by induced electric field is not sufficient to keep the electron in the same orbit, and the radius of curvature of its trajectory will decrease. | ||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | Decrease | ||