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<meta name="description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
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<meta property="og:description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<title>Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?</title>
$2.3.38.$ Potential energy of electrostatic interaction of point charges $q$ and $Q$ located at a distance $r$ from each other, $U = \frac{kqQ}{r}$. Find the electrostatic force. Which charges have repulsion and which have attraction?
</p>
<h3>Solution</h3>
<p>
Let us find the derivative of the potential energy of interaction by distance
$$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$
$$dU=-\frac{kqQ}{r^2}dr\quad(1)$$
Also by the law of conservation of energy, the change in that energy is equal to the work to be done
$$dU=dA\Rightarrow dU=-Fdr\quad(2)$$
<i>NO: The larger $r$ is, the smaller $F$ is</i>
</p>
<center>
<figure>
<img src="draw.png"
loading="lazy" width="270" />
<figcaption>
Forces acting on the system
</figcaption>
</figure>
</center>
<p>
According to Newton's third law
$$\vec{F}_1=-\vec{F}_2=F$$
Substituting $(2)$ into $(1)$ leads to
$$-\frac{kqQ}{r^2}dr=-Fdr$$
$$\boxed{F=\frac{kqQ}{r^2}}$$
When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
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<meta name="description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<meta name="description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<meta property="og:title" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<meta property="og:title" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<meta property="og:image" content="img/logo.png">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<meta property="og:description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?">
<title>Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?</title>
<title>Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?</title>
$2.3.38.$ Potential energy of electrostatic interaction of point charges $q$ and $Q$ located at a distance $r$ from each other, $U = \frac{kqQ}{r}$. Find the electrostatic force. Which charges have repulsion and which have attraction?
$2.3.38.$ Potential energy of electrostatic interaction of point charges $q$ and $Q$ located at a distance $r$ from each other, $U = \frac{kqQ}{r}$. Find the electrostatic force. Which charges have repulsion and which have attraction?
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let us find the derivative of the potential energy of interaction by distance
Let us find the derivative of the potential energy of interaction by distance
$$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$
$$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$
$$dU=-\frac{kqQ}{r^2}dr\quad(1)$$
$$dU=-\frac{kqQ}{r^2}dr\quad(1)$$
Also by the law of conservation of energy, the change in that energy is equal to the work to be done
Also by the law of conservation of energy, the change in that energy is equal to the work to be done
$$dU=dA\Rightarrow dU=-Fdr\quad(2)$$
$$dU=dA\Rightarrow dU=-Fdr\quad(2)$$
<i>NO: The larger $r$ is, the smaller $F$ is</i>
<i>NO: The larger $r$ is, the smaller $F$ is</i>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw.png"
<img src="draw.png"
loading="lazy" width="270" />
loading="lazy" width="270" />
<figcaption>
<figcaption>
Forces acting on the system
Forces acting on the system
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
According to Newton's third law
According to Newton's third law
$$\vec{F}_1=-\vec{F}_2=F$$
$$\vec{F}_1=-\vec{F}_2=F$$
Substituting $(2)$ into $(1)$ leads to
Substituting $(2)$ into $(1)$ leads to
$$-\frac{kqQ}{r^2}dr=-Fdr$$
$$-\frac{kqQ}{r^2}dr=-Fdr$$
$$\boxed{F=\frac{kqQ}{r^2}}$$
$$\boxed{F=\frac{kqQ}{r^2}}$$
When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction.
When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
