Capacitor#

Capacitance#

Capacitance is defined as the constant of proportionality between charge and electric potential.

\[ C \equiv \frac{q}{V} \]

  • \(q\) : Charge of the positive terminal

  • \(V\) : The electric potential required to move a charge from the negative terminal to the positive terminal.

Despite the capacitance is defined with dependence on \(q\) and \(V\), when describing the capacitance of a specific capacitor this is false as any change in \(q\) also changes \(V\) so that \(C\) stays constant. The capacitance is purely dependent on the geometry of the capacitor.

Infinite Sheet#

An infinite plate of an electric field uniformly distributed with electric density \(\rho\) produces a electric field of \(\boldsymbol{E}\) pointinng normal to the surface on both sides of the slate. Using the Gaussian pill box for area \(A\) at a negligible height the enclosed charge is,

\[ Q_\mathrm{enc} = \sigma A \]

The surface integral of the electric field is,

\[ \oint \boldsymbol{E} \cdot \mathrm d \boldsymbol A = 2\boldsymbol{E}A \]

Thus by Gauss’s law,

\[ \boxed{E = \frac{\sigma}{2\epsilon_0}} \]

Infinite Twin Plate Capacitor#

Consider two plate of metal similar to the infinite sheet however with a definite area \(A\). Fill on plate with purely positive charge \(q=Q\) and the other plate purely negative charges \(q=-Q\). We only pay attention to the electric field between the positive plate and the negative plate thus the LHS of Gauss’s law is half that of the infinite sheet,

\[\begin{split} \oint \boldsymbol{E} \cdot \mathrm d \boldsymbol{A} = EA = \frac{Q}{\epsilon_0}\\ E = \frac{Q}{A} \end{split}\]

Let the area be much greater than the separation of the two plates \(A \gg d\) such that the charge are uniformally distributed along the plate with charge density \(\sigma = Q/A\). Thus the electric field is twice that of the infinite sheet.

\[ E = \frac{\sigma}{\epsilon_0} \]

Because the electric field is constant along the uniform sheet, the electric potential is just

\[\begin{split} V = \frac{\sigma}{\epsilon_0}d\\ \boxed{C = \frac{Q}{V} = \frac{A\epsilon_0}{d}} \end{split}\]