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  <h4>Statement</h4>
  <p>
    $6.5.2$
    Two electroconductor pistons of cross section $S$ form a plane capacitor in a tube of an isolating material which contains air at atmospheric pressure $P_0$. How changes the distance between pistons if charges of different signs $\pm Q$ are applied to them? The system is good heat conductor and there is no friction.
  </p>
  <center>
    <figure>
      <img src="statement.png"
        loading="lazy" alt="6.5.2" width="200" />
      <figcaption>
        For problem 6.5.2
      </figcaption>
    </figure>
  </center>
  <h4>Solution</h4>
  <p class="TxtSolutions">
    Let's consider the following figure
  </p>
  <center>
    <figure>
      <img src="draw.png"
        loading="lazy" alt="6.5.2" width="450" />
      <figcaption>
        Forces over pistons
      </figcaption>
    </figure>
  </center>
  <p class="TxtSolutions">
    Applying Newton Second Law over one of the piston, considering the pistons are massless,
  </p>
  <center>
    $F_p = F_{p0} + F_e$
  </center>
  <p class="TxtSolutions">
    where $F_p$, $F_{p0}$ and $F_e$ are forces due to enclosed air between pistons, due to atmospheric pressure and due to electric field, respectively.
  </p>
  <center>
    $P = P_0 + \frac{F_e}{S} \\;(1)$
  </center>
  <p class="TxtSolutions">
    The electric force $F_e$ is defined from the electric field generated by a charged plate that is $E = \frac{\sigma}{2\varepsilon_0}=\frac{Q}{2\varepsilon_0 S}$, so
  </p>
  <center>
    $F_e = Q E =\frac{Q^2}{2\varepsilon_0 S} \\;(2)$
  </center>
  <p class="TxtSolutions">
    Putting $(2)$ into $(1)$
  </p>
  <center>
    $P = P_0 + \frac{Q^2}{2\varepsilon_0 S} \\;(3)$
  </center>
  <p class="TxtSolutions">
    Taking account that the system is good heat conductor, we can suppose that thermal equilibrium is instantaneously established or that the container acts as a thermos, keeping the temperature unchanged.Then, applying Boyle-Mariotte law
  </p>
  <center>
    $P_0 S L = P S l$
  </center>
  <center>
    $\frac{L}{l} = \frac{P}{P_0} \\;(4)$
  </center>
  <p class="TxtSolutions">
    Substituting $(3)$ into $(4)$
   </p>
    <center>
      $\boxed{\frac{L}{l} = 1 + \frac{Q^2}{2P_0\varepsilon_0 S^2}}$
    </center>
    <p class="TxtSolutions" style="text-align: center; font-style: italic; font-size: 14;">   
        BSc. Luis Daniel Fernández Quintana<br>
        Physics Department (FCNE)<br>
        Universidad de Oriente, Cuba<br>
    </p>
</article>