<h3 id="back-link"><a href="/#7.1">$\leftarrow$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>
      <figure>
        <img src="statement.png"
          loading="lazy" alt="7.1.6" width="300" />
        <figcaption>
          For problem 7.1.6
        </figcaption>
      </figure>
    </center>

    <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
      $$l = v_0\cos{\alpha}\~t_v \\;(4)$$
      The time expended by particles during the path through the capacitor can be found from acceleration definition
      $$\vec{a_y} = \frac{\Delta \vec{v_y}}{\Delta t} = \frac{\vec{v_{1y}}-\vec{v_{0y}}}{t_v}$$
      $$a_y = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{t_v}$$
      $$t_v = \frac{v_1\sin{\beta}+v_0\sin{\alpha}}{a_y} \\;(5)$$
      Substituting $v_1$ from $(3)$ into $(5)$ and developing algebraically
      $$t_v = v_0\cos{\alpha}\frac{\tan{\alpha}+\tan{\beta}}{a_y} \\;(6)$$
      Putting $(6)$ into $(4)$ and separating $v_0$
      $$v_0^2 = \frac{a_yl}{\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}$$
      and according to $(2)$,
      $$v_0^2 = \frac{qEl}{m\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})} \\;(7)$$
      Finally, substituting $(7)$ into $(1)$
      $$\boxed{K_0 = \frac{qEl}{2\cos^2{\alpha}(\tan{\alpha}+\tan{\beta})}}$$
    </p>


    <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>
    </p>



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