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+ <header style="text-align:center;">
<h2>Solutions of Savchenko Physics Textbook</h2>
<p class="author">
Aliaksandr Melnichenka <br/>
October 2023
</p>
</header>
− <main>
− <article>
− <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
−
− <h4>Statement</h4>
− <p>
− $7.1.6$
− Particles with mass $m$ and charge $q$ enter in a capacitor of length $l$ under an angle $\alpha$ respect to plate's plane and exit forming an angle $\beta$. Find the initial kinetic energy of particles, if the field intensity inside the capacitor is $E$.
− </p>
− <center>
− <figure>
− <img src="statement.png"
− loading="lazy" alt="7.1.6" width="300" />
− <figcaption>
− For problem 7.1.6
− </figcaption>
− </figure>
− </center>
− <h4>Solution</h4>
− <p class="TxtSolutions">
− Initial kinetic energy is defined as
− </p>
− <center>
− $K_0 = \frac{mv_0^2}{2} \;(1)$
− </center>
− <p class="TxtSolutions">
− If we apply Newton's Second Law, acceleration can be defined,
− </p>
− <center>
− $a_y = \frac{qE}{m} \;(2)$
− </center>
− <p class="TxtSolutions">
− As there are no forces in horizontal direction, the component of velocity in that direction is unaffected
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $7.1.6$
+ Particles with mass $m$ and charge $q$ enter in a capacitor of length $l$ under an angle $\alpha$ respect to plate's plane and exit forming an angle $\beta$. Find the initial kinetic energy of particles, if the field intensity inside the capacitor is $E$.
</p>
<center>
− $v_0\cos{\alpha}=v_1\cos{\beta} \;(3)$
+ <figure>
+ <img src="statement.png"
+ loading="lazy" alt="7.1.6" width="300" />
+ <figcaption>
+ For problem 7.1.6
+ </figcaption>
+ </figure>
</center>
− <p class="TxtSolutions">
+
+ <h3>Solution</h3>
+ <p>
+ Initial kinetic energy is defined as
+ $$K_0 = \frac{mv_0^2}{2} \;(1)$$
+ If we apply Newton's Second Law, acceleration can be defined,
+ $$a_y = \frac{qE}{m} \;(2)$$
+ As there are no forces in horizontal direction, the component of velocity in that direction is unaffected
+ $$v_0\cos{\alpha}=v_1\cos{\beta} \;(3)$$
For the horizontal path until exit's point
− </p>
− <center>
− $l = v_0\cos{\alpha}~t_v \;(4)$
− </center>
− <p class="TxtSolutions">
+ $$l = v_0\cos{\alpha}~t_v \;(4)$$
The time expended by particles during the path through the capacitor can be found from acceleration definition
− </p>
− <center>
− $\vec{a_y} = \frac{\Delta \vec{v_y}}{\Delta t} = \frac{\vec{v_{1y}}-\vec{v_{0y}}}{t_v}$
− </center>
− <center>
− $a_y = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{t_v}$
− </center>
− <center>
− $t_v = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{a_y} \;(5)$
− </center>
− <p clas="TxtSolutions">
+ $$\vec{a_y} = \frac{\Delta \vec{v_y}}{\Delta t} = \frac{\vec{v_{1y}}-\vec{v_{0y}}}{t_v}$$
+ $$a_y = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{t_v}$$
+ $$t_v = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{a_y} \;(5)$$
Substituting $v_1$ from $(3)$ into $(5)$ and developing algebraically
− </p>
− <center>
− $t_v = v_0\cos{\alpha}\frac{\tan{\alpha}+\tan{\beta}}{a_y} \;(6)$
− </center>
− <p class="TxtSolutions">
+ $$t_v = v_0\cos{\alpha}\frac{\tan{\alpha}+\tan{\beta}}{a_y} \;(6)$$
Putting $(6)$ into $(4)$ and separating $v_0$
− </p>
− <center>
− $v_0^2 = \frac{a_yl}{\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}$
− </center>
− <p class="TxtSolutions">
+ $$v_0^2 = \frac{a_yl}{\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}$$
and according to $(2)$,
− </p>
− <center>
− $v_0^2 = \frac{qEl}{m\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})} \;(7)$
− </center>
− <p class="=TxtSolutions">
+ $$v_0^2 = \frac{qEl}{m\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})} \;(7)$$
Finally, substituting $(7)$ into $(1)$
+ $$\boxed{K_0 = \frac{qEl}{2\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}}$$
</p>
− <center>
− $\boxed{K_0 = \frac{qEl}{2\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}}$
− </center>
− <p class="TxtSolutions" style="text-align: right; font-style: italic; font-size: 14;">
− BSc. Luis Daniel Fernández Quintana<br>
− Physics Department (FCNE)<br>
− Universidad de Oriente, Cuba<br>
− </p>
− </article>
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+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ BSc. Luis Daniel Fernández Quintana<br>
+ Physics Department (FCNE)<br>
+ Universidad de Oriente, Cuba<br>
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