Новое решение
en/11.2.21.md
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| + | ### Statement | ||
| + | |||
| + | $11.2.21.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
| + | |||
| + | We model the electron as a sphere of radius R with charge e distributed on its surface. The electrostatic potential energy in SI is: | ||
| + | |||
| + | $U = \frac{1}{2} \cdot \frac{1}{4\pi\varepsilon_0} \cdot \frac{e^2}{R}$ | ||
| + | |||
| + | The equivalent electromagnetic mass, according to $E = mc^2 $ is | ||
| + | |||
| + | $m_{\text{em}} = \frac{U}{c^2}$ | ||
| + | |||
| + | Data | ||
| + | |||
| + | Electron charge: $e = 1.60 \times 10^{-19} \text{ C}$ | ||
| + | Coulomb constant: $\frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9 \, \text{N·m}^2/\text{C}^2$ | ||
| + | |||
| + | Radius (experimental limit): $R = 10^{-18} \text{ m}$ | ||
| + | |||
| + | Speed of light: $c = 3.0 \times 10^8 \text{ m/s}$ | ||
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| + | Calculation | ||
| + | |||
| + | $e^2 = (1.60 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 | ||
| + | \frac{1}{4\pi\varepsilon_0} e^2 = (9.0 \times 10^9) \times (2.56 \times 10^{-38}) = 2.304 \times 10^{-28} \, \text{J·m}$ | ||
| + | |||
| + | Potential energy U: | ||
| + | |||
| + | $U = \frac{1}{2} \times \frac{2.304 \times 10^{-28}}{10^{-18}} = \frac{1}{2} \times 2.304 \times 10^{-10} = 1.152 \times 10^{-10} \, \text{J}$ | ||
| + | |||
| + | Electromagnetic mass: | ||
| + | |||
| + | $m_{\text{em}} = \frac{U}{c^2} = \frac{1.152 \times 10^{-10}}{(3.0 \times 10^8)^2} = \frac{1.152 \times 10^{-10}}{9.0 \times 10^{16}} = 1.28 \times 10^{-27} \, \text{kg}$ | ||
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| + | Result | ||
| + | |||
| + | $\boxed{m_{\text{em}} \approx 1.3 \times 10^{-27} \text{ kg}}$ | ||
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| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $11.2.21.$ [Insert the problem statement] | |||
| ### Solution | |||
| We model the electron as a sphere of radius R with charge e distributed on its surface. The electrostatic potential energy in SI is: | |||
| $U = \frac{1}{2} \cdot \frac{1}{4\pi\varepsilon_0} \cdot \frac{e^2}{R}$ | |||
| The equivalent electromagnetic mass, according to $E = mc^2 $ is | |||
| $m_{\text{em}} = \frac{U}{c^2}$ | |||
| Data | |||
| Electron charge: $e = 1.60 \times 10^{-19} \text{ C}$ | |||
| Coulomb constant: $\frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9 \, \text{N·m}^2/\text{C}^2$ | |||
| Radius (experimental limit): $R = 10^{-18} \text{ m}$ | |||
| Speed of light: $c = 3.0 \times 10^8 \text{ m/s}$ | |||
| Calculation | |||
| $e^2 = (1.60 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 | |||
| \frac{1}{4\pi\varepsilon_0} e^2 = (9.0 \times 10^9) \times (2.56 \times 10^{-38}) = 2.304 \times 10^{-28} \, \text{J·m}$ | |||
| Potential energy U: | |||
| $U = \frac{1}{2} \times \frac{2.304 \times 10^{-28}}{10^{-18}} = \frac{1}{2} \times 2.304 \times 10^{-10} = 1.152 \times 10^{-10} \, \text{J}$ | |||
| Electromagnetic mass: | |||
| $m_{\text{em}} = \frac{U}{c^2} = \frac{1.152 \times 10^{-10}}{(3.0 \times 10^8)^2} = \frac{1.152 \times 10^{-10}}{9.0 \times 10^{16}} = 1.28 \times 10^{-27} \, \text{kg}$ | |||
| Result | |||
| $\boxed{m_{\text{em}} \approx 1.3 \times 10^{-27} \text{ kg}}$ | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||