Правка разделов «Problem», «Solution», «Answer»
en/7.2.11.md
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| − | ### | ||
| + | ### Problem | ||
| − | $7.2.11.$ [Insert the problem statement] | ||
| + | $7.2.11^*.$ Solve problem 7.2.10 in the case when a beam of electrons emitted from a point located at a distance $L \gg R$ from the center of the spheres makes a small angle with the normal to its surface. | ||
| ### Solution | |||
| − | |||
| + | <b>1. Spherical symmetry and optical axis</b>\ | ||
| + | The system of concentric spheres possesses perfect spherical symmetry. This means it does not have a single designated optical axis like a conventional glass lens. Any straight line passing through the point source and the center of the spheres can be considered as the principal optical axis of the system. | ||
| − | #### Answer | ||
| + | <b>2. Paraxial approximation</b>\ | ||
| + | The condition that the electron beam makes a small angle with the normal to the surface of the sphere (i.e., with the radius vector) means that the rays propagate at very small angles to our chosen optical axis. In optics, this is called a paraxial beam. Under these conditions, spherical aberration can be neglected, and the system will focus the rays into one sharp focal point. | ||
| − | |||
| + | <b>3. Thin lens formula</b>\ | ||
| + | As proven in problem [7.2.10](https://savchenkosolutions.com/en/7.2.10), this system of spheres overall acts as a converging electrostatic lens located at its center. | ||
| + | Let us apply the standard thin lens formula for paraxial rays: | ||
| + | $$ \frac{1}{L} + \frac{1}{x} = \frac{1}{f} $$ | ||
| + | where $L$ is the distance from the source to the lens (the center of the spheres), $x$ is the required distance to the focusing point, and $f$ is the focal length of the system of spheres. | ||
| + | |||
| + | Let us express the required distance $x$: | ||
| + | $$ \frac{1}{x} = \frac{1}{f} - \frac{1}{L} \implies x = \frac{1}{\frac{1}{f} - \frac{1}{L}} $$ | ||
| + | |||
| + | The value of the focal length $f$ is substituted from the results of problem 7.2.10. | ||
| + | |||
| + | <i>Note: The formula is universal. Depending on the chosen physical model of the holes in problem 7.2.10, you can substitute either the linear value $f = R(V_0/V)$ (for ideal grids) or the quadratic value $f = 2R(V_0/V)^2$ (for open apertures). In the official answer, the authors of the textbook use the quadratic model.</i> | ||
| + | |||
| + | #### Answer | ||
| + | $$ x \approx \frac{1}{1/f - 1/L} $$ | ||
| + | where $f$ is the focal length found in problem 7.2.10 (in the authors' model $f = 2R(V_0/V)^2$). | ||
| @@ -1,11 +1,28 @@ | |||
| ### |
### Problem | ||
| $7.2.11.$ [Insert the problem statement] | $7.2.11^*.$ Solve problem 7.2.10 in the case when a beam of electrons emitted from a point located at a distance $L \gg R$ from the center of the spheres makes a small angle with the normal to its surface. | ||
| ### Solution | ### Solution | ||
| <b>1. Spherical symmetry and optical axis</b>\ | |||
| The system of concentric spheres possesses perfect spherical symmetry. This means it does not have a single designated optical axis like a conventional glass lens. Any straight line passing through the point source and the center of the spheres can be considered as the principal optical axis of the system. | |||
| #### Answer | <b>2. Paraxial approximation</b>\ | ||
| The condition that the electron beam makes a small angle with the normal to the surface of the sphere (i.e., with the radius vector) means that the rays propagate at very small angles to our chosen optical axis. In optics, this is called a paraxial beam. Under these conditions, spherical aberration can be neglected, and the system will focus the rays into one sharp focal point. | |||
| <b>3. Thin lens formula</b>\ | |||
| As proven in problem [7.2.10](https://savchenkosolutions.com/en/7.2.10), this system of spheres overall acts as a converging electrostatic lens located at its center. | |||
| Let us apply the standard thin lens formula for paraxial rays: | |||
| $$ \frac{1}{L} + \frac{1}{x} = \frac{1}{f} $$ | |||
| where $L$ is the distance from the source to the lens (the center of the spheres), $x$ is the required distance to the focusing point, and $f$ is the focal length of the system of spheres. | |||
| Let us express the required distance $x$: | |||
| $$ \frac{1}{x} = \frac{1}{f} - \frac{1}{L} \implies x = \frac{1}{\frac{1}{f} - \frac{1}{L}} $$ | |||
| The value of the focal length $f$ is substituted from the results of problem 7.2.10. | |||
| <i>Note: The formula is universal. Depending on the chosen physical model of the holes in problem 7.2.10, you can substitute either the linear value $f = R(V_0/V)$ (for ideal grids) or the quadratic value $f = 2R(V_0/V)^2$ (for open apertures). In the official answer, the authors of the textbook use the quadratic model.</i> | |||
| #### Answer | |||
| $$ x \approx \frac{1}{1/f - 1/L} $$ | |||
| where $f$ is the focal length found in problem 7.2.10 (in the authors' model $f = 2R(V_0/V)^2$). | |||