# 8.5 Molecular model of a dielectric  (Page 5/13)

 Page 5 / 13

## Key equations

 Capacitance $C=\frac{Q}{V}$ Capacitance of a parallel-plate capacitor $C={\epsilon }_{0}\frac{A}{d}$ Capacitance of a vacuum spherical capacitor $C=4\pi {\epsilon }_{0}\frac{{R}_{1}{R}_{2}}{{R}_{2}-{R}_{1}}$ Capacitance of a vacuum cylindrical capacitor $C=\frac{2\pi {\epsilon }_{0}l}{\text{ln}\left({R}_{2}\text{/}{R}_{1}\right)}$ Capacitance of a series combination $\frac{1}{{C}_{\text{S}}}=\frac{1}{{C}_{1}}+\frac{1}{{C}_{2}}+\frac{1}{{C}_{3}}+\text{⋯}$ Capacitance of a parallel combination ${C}_{\text{P}}={C}_{1}+{C}_{2}+{C}_{3}+\text{⋯}$ Energy density ${u}_{E}=\frac{1}{2}{\epsilon }_{0}{E}^{2}$ Energy stored in a capacitor ${U}_{C}=\frac{1}{2}{V}^{2}C=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\frac{{Q}^{2}}{C}=\frac{1}{2}QV$ Capacitance of a capacitor with dielectric $C=\kappa {C}_{0}$ Energy stored in an isolated capacitor with dielectric $U=\frac{1}{\kappa }{U}_{0}$ Dielectric constant $\kappa =\frac{{E}_{0}}{E}$ Induced electrical field in a dielectric ${\stackrel{\to }{E}}_{\text{i}}=\left(\frac{1}{\kappa }-1\right){\stackrel{\to }{E}}_{0}$

## Conceptual questions

Distinguish between dielectric strength and dielectric constant.

Dielectric strength is a critical value of an electrical field above which an insulator starts to conduct; a dielectric constant is the ratio of the electrical field in vacuum to the net electrical field in a material.

Water is a good solvent because it has a high dielectric constant. Explain.

Water has a high dielectric constant. Explain why it is then not used as a dielectric material in capacitors.

Water is a good solvent.

Elaborate on why molecules in a dielectric material experience net forces on them in a non-uniform electrical field but not in a uniform field.

Explain why the dielectric constant of a substance containing permanent molecular electric dipoles decreases with increasing temperature.

When energy of thermal motion is large (high temperature), an electrical field must be large too in order to keep electric dipoles aligned with it.

Give a reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. How does a dielectric material also allow a greater voltage to be applied to a capacitor? (The dielectric thus increases C and permits a greater V .)

Elaborate on the way in which the polar character of water molecules helps to explain water’s relatively large dielectric constant.

Sparks will occur between the plates of an air-filled capacitor at a lower voltage when the air is humid than when it is dry. Discuss why, considering the polar character of water molecules.

## Problems

Two flat plates containing equal and opposite charges are separated by material 4.0 mm thick with a dielectric constant of 5.0. If the electrical field in the dielectric is 1.5 MV/m, what are (a) the charge density on the capacitor plates, and (b) the induced charge density on the surfaces of the dielectric?

For a Teflon™-filled, parallel-plate capacitor, the area of the plate is $50.0\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ and the spacing between the plates is 0.50 mm. If the capacitor is connected to a 200-V battery, find (a) the free charge on the capacitor plates, (b) the electrical field in the dielectric, and (c) the induced charge on the dielectric surfaces.

a. 37 nC; b. 0.4 MV/m; c. 19 nC

Find the capacitance of a parallel-plate capacitor having plates with a surface area of $5.00\phantom{\rule{0.2em}{0ex}}{m}^{2}$ and separated by 0.100 mm of Teflon™.

(a) What is the capacitance of a parallel-plate capacitor with plates of area $1.50\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?

a. $4.4\phantom{\rule{0.2em}{0ex}}\text{μ}\text{F}$ ; b. $4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\text{-5}}\phantom{\rule{0.2em}{0ex}}\text{C}$

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