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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $2.3.39.$ One research institute decided to use an expression for the potential energy of point charges in the form $U' = \frac{kqQ}{r} - \frac{kqQ}{R}$, where $R$ is a constant distance set once and for all. Will using $U'$ instead of $U = \frac{kqQ}{r}$ affect the results of particle motion calculations? | | $2.3.39.$ One research institute decided to use an expression for the potential energy of point charges in the form $U' = \frac{kqQ}{r} - \frac{kqQ}{R}$, where $R$ is a constant distance set once and for all. Will using $U'$ instead of $U = \frac{kqQ}{r}$ affect the results of particle motion calculations? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Expression for the potential energy | | Expression for the potential energy |
| $$ | | $$ |
| U = \frac{kqQ}{r} | | U = \frac{kqQ}{r} |
| $$ | | $$ |
| | | |
| $$ | | $$ |
| U' = \frac{kqQ}{r} - \frac{kqQ}{R} | | U' = \frac{kqQ}{r} - \frac{kqQ}{R} |
| $$ | | $$ |
| | | |
| By statement, $R$ is a constant distance | | By statement, $R$ is a constant distance |
| $$ | | $$ |
| R = \text{const} | | R = \text{const} |
| $$ | | $$ |
| Derivatives for potential energy of expressions | | Derivatives for potential energy of expressions |
| $$ | | $$ |
| (U_x)' = p | | (U_x)' = p |
| $$ | | $$ |
| | | |
| $$ | | $$ |
| (U)' = - \frac{kqQ}{r^2} \, dr | | (U)' = - \frac{kqQ}{r^2} \, dr |
| $$ | | $$ |
| | | |
| $$ | | $$ |
| (U')' = - \frac{kqQ}{r^2} \, dr | | (U')' = - \frac{kqQ}{r^2} \, dr |
| $$ | | $$ |
| | | |
| The derivative of the constant is zero | | The derivative of the constant is zero |
| $$ | | $$ |
| (\text{const})' = 0 \Rightarrow (R)' = 0 | | (\text{const})' = 0 \Rightarrow (R)' = 0 |
| $$ | | $$ |
| | | |
| $$ | | $$ |
| (U)' = (U')'! | | (U)' = (U')'! |
| $$ | | $$ |
| Thus, the speeds remain unchanged and the motion of the particles is not affected | | Thus, the speeds remain unchanged and the motion of the particles is not affected |
| $$ | | $$ |
| V = V'! | | V = V'! |
| $$ | | $$ |
| | | |
| | | |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $ | | $ |
| \text{No, it won't!} | | \text{No, it won't!} |
| $ | | $ |
| </p> | | </p> |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
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