Question 16.8: A parallel-plate capacitor has square plates 1.1 cm on a sid...
A parallel-plate capacitor has square plates 1.1 cm on a side separated by 0.15 mm of Teflon. Find its capacitance and the maximum potential difference that it can sustain.
ORGANIZE AND PLAN Capacitance depends on two things: the capacitor’s dimensions (Section 16.4) and the dielectric. The maximum potential difference depends on plate spacing and the maximum electric field; the latter follows from the dielectric strength.
Without a dielectric, the capacitance of a parallel-plate capacitor is C=\varepsilon_{0} A / d , with A the plate area and d the separation. Inserting the dielectric (Figure 16.23) increases the capacitance by a factor of κ , so C=\kappa C_{0}=\kappa \varepsilon_{0} A / d. To find the maximum potential difference, recall V = Ed that for the uniform field in a parallel-plate capacitor.
So the maximum V is V \text { is } V_{\max }=E_{\max } d \text {, where } E_{\max } is the dielectric strength.
\text { Known: } A=(1.1 cm )^{2} ; d=0.15 mm .
C=\kappa \frac{\varepsilon_{0} A}{d}=(2.1) \frac{\left(8.85 \times 10^{-12} F / m \right)(0.011 m )^{2}}{1.5 \times 10^{-4} m }.
=1.5 \times 10^{-11} F =15 pF.
Table 16.3 gives 60 MV/m for the dielectric strength of Teflon, so the maximum potential difference for our capacitor is
V_{\max }=E_{\max } d=\left(60 \times 10^{6} V / m \right)\left(1.5 \times 10^{-4} m \right).
=9.0 \times 10^{3} V =9.0 kV.
| TABLE 16.3 Dielectric Properties of Selected Materials (Measured at 20°C) | ||
| Dielectric strength E_{\max }( MV / m ) | Dielectric constant κ |
Material |
| – | 1 (exact) | Vacuum |
| 3.0 | 1.00058 | Air |
| 60 | 2.1 | Teflon |
| 25 | 2.6 | Polystyrene |
| 14 | 3.4 | Nylon |
| 16 | 3.7 | Paper |
| 14 | 5.6 | Glass (Pyrex) |
| 12 | 6.7 | Neoprene |
| 500 | 26 | Tantalum oxide |
| Depends on purity | 80 | Water |
| 8.0 | 256 | Strontium titanate |
REFLECT That 0.15-mm plate spacing is actually fairly large, and results in a small capacitance C. But at the same time it gives a large V_{\max }. As in this example, it’s easy to make a small C with a large V_{\max } And it’s easy to make a large C by decreasing the plate spacing -at the cost of reducing V_{\max } . What’s difficult, and therefore expensive, is to make a capacitor with both a large capacitance and a large maximum potential difference.