<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta property="og:title" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<title>One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?</title>
$2.3.39.$ 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>
<h3>Solution</h3>
<p>
Expression for the potential energy
$$
U = \frac{kqQ}{r}
$$
$$
U' = \frac{kqQ}{r} - \frac{kqQ}{R}
$$
By statement, $R$ is a constant distance
$$
R = \text{const}
$$
Derivatives for potential energy of expressions
$$
(U_x)' = p
$$
$$
(U)' = - \frac{kqQ}{r^2} \, dr
$$
$$
(U')' = - \frac{kqQ}{r^2} \, dr
$$
The derivative of the constant is zero
$$
(\text{const})' = 0 \Rightarrow (R)' = 0
$$
$$
(U)' = (U')'!
$$
Thus, the speeds remain unchanged and the motion of the particles is not affected
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta name="description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta property="og:title" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta property="og:title" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<meta property="og:description" content="One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?">
<title>One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?</title>
<title>One research institute decided to use an expression for the potential energy of point charges in the form U' = rac{kqQ}{r} - rac{kqQ}{R}, where R is a constant distance set once and for all. Will using U' instead of U = rac{kqQ}{r} affect the results of particle motion calculations?</title>
$2.3.39.$ 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
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
