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<title>A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.</title>
$7.3.9^*.$ A thin electron beam accelerated by the potential difference $V$ enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage $V_0$ sin wt is applied to the capacitor plates. The distance between the plates of the capacitor $d$ is much smaller than its length $l$.
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<center>
<figure>
<img src="statement.png"
loading="lazy" width="230" />
<figcaption>
For problem $7.3.9^*$
</figcaption>
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<p>
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<h3>Solution</h3>
<p>
</p>
<center>
<figure>
<img src="draw.png"
loading="lazy" width="230" />
<figcaption>
Trajectory of a particle in an electric field
</figcaption>
</figure>
</center>
<p>
From the drawing
$$\tan\alpha = \frac{v_y}{v_x}$$
Law of conservation of energy
$$\frac{mv^2_x}{2}=eU$$
From where
$$v_x=\sqrt{\frac{2eU}{m}}$$
Force $\vec{F}$ acting on the particle:
$$F=eU=e\frac{U_0}{d}\sin\omega t$$
Second Newton's Laws
$$ma=\frac{eU_0}{d}\sin\omega t$$
By the definition of acceleration $a = \frac{dv}{dt}$
<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">
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<meta name="description" content="A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.">
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<meta property="og:description" content="A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.">
<meta property="og:description" content="A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.">
<title>A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.</title>
<title>A thin electron beam accelerated by the potential difference V enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage V_0 sin wt is applied to the capacitor plates. The distance between the plates of the capacitor d is much smaller than its length l.</title>
$7.3.9^*.$ A thin electron beam accelerated by the potential difference $V$ enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage $V_0$ sin wt is applied to the capacitor plates. The distance between the plates of the capacitor $d$ is much smaller than its length $l$.
$7.3.9^*.$ A thin electron beam accelerated by the potential difference $V$ enters a flat capacitor parallel to its plates. Determine the angular spread of electrons if a voltage $V_0$ sin wt is applied to the capacitor plates. The distance between the plates of the capacitor $d$ is much smaller than its length $l$.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
For problem $7.3.9^*$
For problem $7.3.9^*$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw.png"
<img src="draw.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
Trajectory of a particle in an electric field
Trajectory of a particle in an electric field
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
From the drawing
From the drawing
$$\tan\alpha = \frac{v_y}{v_x}$$
$$\tan\alpha = \frac{v_y}{v_x}$$
Law of conservation of energy
Law of conservation of energy
$$\frac{mv^2_x}{2}=eU$$
$$\frac{mv^2_x}{2}=eU$$
From where
From where
$$v_x=\sqrt{\frac{2eU}{m}}$$
$$v_x=\sqrt{\frac{2eU}{m}}$$
Force $\vec{F}$ acting on the particle:
Force $\vec{F}$ acting on the particle:
$$F=eU=e\frac{U_0}{d}\sin\omega t$$
$$F=eU=e\frac{U_0}{d}\sin\omega t$$
Second Newton's Laws
Second Newton's Laws
$$ma=\frac{eU_0}{d}\sin\omega t$$
$$ma=\frac{eU_0}{d}\sin\omega t$$
By the definition of acceleration $a = \frac{dv}{dt}$
By the definition of acceleration $a = \frac{dv}{dt}$
<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>
