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

    <h3> Statement </h3>
    <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>

    <h3>Solution</h3>
    <p>
      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>
      Applying Newton Second Law over one of the piston, considering the pistons are massless,
      $$F_p = F_{p0} + F_e$$
      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.
      $$P = P_0 + \frac{F_e}{S} \\;(1)$$
      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
      $$F_e = Q E =\frac{Q^2}{2\varepsilon_0 S} \\;(2)$$
      Putting $(2)$ into $(1)$
      $$P = P_0 + \frac{Q^2}{2\varepsilon_0 S} \\;(3)$$
      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
      $$P_0 S L = P S l$$
      $$\frac{L}{l} = \frac{P}{P_0} \\;(4)$$
      Substituting $(3)$ into $(4)$
      $$\boxed{\frac{L}{l} = 1 + \frac{Q^2}{2P_0\varepsilon_0 S^2}}$$
    </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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