Translated 9.2.1-9.2.9

astrosander edited Oct 17, 2024, 12:51 AM

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		+	<title>An electric field of strength E~ of charges moving with velocity v creates a magnetic field whose induction \vec{B} = K[\vec{v}\times \vec{E}]. The coefficient K is equal to \varepsilon_0 \mu_0in SI and 1/c in CGS, where c is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.</title>
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		+	<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
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		+	<p class="author">
		+	Solutions of Savchenko Problems in Physics <br>
		+	<i><b>knowledge must be free</b></i>
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		+	<h3 id="back-link"><a href="../../#9.2">$\leftarrow$Back</a></h3>
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		+	<h3> Statement </h3>
		+	<p>
		+	$9.2.1.$ An electric field of strength $E~$ of charges moving with velocity $v$ creates a magnetic field whose induction $\vec{B} = K[\vec{v}\times \vec{E}]$. The coefficient $K$ is equal to $\varepsilon_0 \mu_0$
		+	in SI and $1/c$ in CGS, where $c$ is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/9/9.2.1/statement.png"
		+	loading="lazy" width="150" />
		+	<figcaption>
		+	For problem $9.2.1$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Proof</h3>
		+	<p>
		+	Magnetic field, the induction of which
		+	$$ \|\vec{B}\| = \varepsilon_0 \mu_0\left[\vec{v}\times \vec{E}\right] \le \varepsilon_0 \mu_0vE $$
		+	then the force of magnetic interaction (Lorentz force) of charges
		+	$$ F_B = \|q[\vec{v} \times \vec{B}]\| = qvB \le \varepsilon_0 \mu_0 qv^2E $$
		+	taking into account the force of electrostatic interaction $F_E = qE$, we obtain
		+	$$ \frac{F_B}{F_E} \le \varepsilon_0 \mu_0 v^2 = \frac{v^2}{c^2} $$
		+	Considering the smallness of the speed $v \ll c$, then
		+	$$ \boxed{\frac{F_B}{F_E} \approx 0} $$
		+	</p>
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Andrei Yersh<br>
		+	</p>
		+
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			<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
			<meta name="description" content="An electric field of strength E~ of charges moving with velocity v creates a magnetic field whose induction \vec{B} = K[\vec{v}\times \vec{E}]. The coefficient K is equal to \varepsilon_0 \mu_0in SI and 1/c in CGS, where c is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="An electric field of strength E~ of charges moving with velocity v creates a magnetic field whose induction \vec{B} = K[\vec{v}\times \vec{E}]. The coefficient K is equal to \varepsilon_0 \mu_0in SI and 1/c in CGS, where c is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.">
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			<meta property="og:description" content="An electric field of strength E~ of charges moving with velocity v creates a magnetic field whose induction \vec{B} = K[\vec{v}\times \vec{E}]. The coefficient K is equal to \varepsilon_0 \mu_0in SI and 1/c in CGS, where c is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>An electric field of strength E~ of charges moving with velocity v creates a magnetic field whose induction \vec{B} = K[\vec{v}\times \vec{E}]. The coefficient K is equal to \varepsilon_0 \mu_0in SI and 1/c in CGS, where c is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.</title>
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			<header style="text-align:center;">
			<a href="../../" style="text-decoration: none;">
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			<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
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			Solutions of Savchenko Problems in Physics <br>
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			</p>
			</header>

			<h3 id="back-link"><a href="../../#9.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$9.2.1.$ An electric field of strength $E~$ of charges moving with velocity $v$ creates a magnetic field whose induction $\vec{B} = K[\vec{v}\times \vec{E}]$. The coefficient $K$ is equal to $\varepsilon_0 \mu_0$
			in SI and $1/c$ in CGS, where $c$ is the speed of light. Prove that the magnetic interaction of two moving charges is weaker than their electrical interaction.
			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/9/9.2.1/statement.png"
			loading="lazy" width="150" />
			<figcaption>
			For problem $9.2.1$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Proof</h3>
			<p>
			Magnetic field, the induction of which
			$$ \|\vec{B}\| = \varepsilon_0 \mu_0\left[\vec{v}\times \vec{E}\right] \le \varepsilon_0 \mu_0vE $$
			then the force of magnetic interaction (Lorentz force) of charges
			$$ F_B = \|q[\vec{v} \times \vec{B}]\| = qvB \le \varepsilon_0 \mu_0 qv^2E $$
			taking into account the force of electrostatic interaction $F_E = qE$, we obtain
			$$ \frac{F_B}{F_E} \le \varepsilon_0 \mu_0 v^2 = \frac{v^2}{c^2} $$
			Considering the smallness of the speed $v \ll c$, then
			$$ \boxed{\frac{F_B}{F_E} \approx 0} $$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Andrei Yersh<br>
			</p>


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