Dielectrics with Capacitors

Let's revisit the calculation of the capacitance of parallel plates. The capacitance is defined as \(C={Q\over\Dl V}\), and we found the potential difference between the plates by integrating the electric field. So $$C=\frac{Q}{\int E_z\,dz}$$ Notice that the capacitance is inversely proportional to the electric field between the plates. If I can make that field smaller, then the capacitance would be bigger.

Now remember what happens when you place an insulator in an electric field? The insulator polarizes, creating a counterfield which partially cancels out the original field, so that the net electric field inside the insulator is smaller than the field outside, by a factor of the dielectric constant \(\kappa\).

Repeat the calculation in Example 9.5.1 to find the capacitance of a parallel-plate capacaitor, where each plate has area \(A\), the plates are a distance \(d\) apart, but the space between the plates is filled with paper, which has a dielectric constant of \(\kappa=3.5\).
Suppose that charge \(+Q\) is on the top plate and charge \(-Q\) is on the bottom plate. As in Example 9.5.1, the plates would create an electric field of \(4\pi kQ/A\) between them if the space wasn't filled with paper. With the paper, the electric field is smaller by a factor of \(\kappa\): \(E=\frac{4\pi kQ}{\kappa A}\). Continuing on with the original calculation, we have that the potential difference between the plates is \(\Dl V=\frac{4\pi kQd}{\kappa A}\), and the capacitance is $$C={Q\over\Dl V}={Q\over 4\pi kQd/\kappa A}=\kappa\frc{4\pi k}{A\over d}$$
Generally, an insulator inserted in between the plates of a capacitor will increase its capacitance by a factor of \(\kappa\). This can be rather significant! Paper has a dielectric constant \(\kappa=3.5\), so it almost quadruples the capacitance. Pure water will increase the capacitance by a factor of 80, and there are other materials with even higher dielectric constants.

Consider, however, that the materials with the highest dielectric constants are the metals: a pure conductor has \(\kappa=\infty\). Of course, we can't insert metal in between the plates of a capacitor: it would immediately discharge. Even if you use an insulator, you must look out for the phenomenon of electric breakdown mentioned in . If you place too much charge on a capacitor, so that the electric field between the plates exceeds the breakdown threshold of the insulator between them (and air counts as an insulator in this case!), then the insulator will become a temporary conductor, and the capacitor will discharge.

To construct a high-capacitance capacitor, then, we would like to find materials which have high dielectric constants and high breakdown thresholds. There are other tricks one can use to construct these ultracapacitors, including creating battery-capacitor hybrids which have the benefits of both.