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Reading: Chapter 24, CAPACITANCE AND DIELECTRICS
Study guide: Chapter 25
Find a relationship between $\Delta V$ and $Q$...
Gausses law says... $Ea=q/\epsilon_0$, then
$$\Delta V=E\cdot d = \frac{\sigma}{\epsilon_0} d = \frac{Qd}{A\epsilon_0}$$
(That is... $Q=EA\epsilon_0$ **)
$$\Rightarrow Q=\frac{A\epsilon_0}{d}\Delta V \equiv
C\Delta V$$
Getting a little sloppy.... the convention is to write the potential difference across the capacitor $\Delta V\equiv$ "$V$", then,$$Q=CV$$
Capacitance $C$ ($=\epsilon_0\frac{A}{d}$ for a flat plate capacitor):
[Capacitance of a 1 cm X 1 cm plate with a gap of 2 mm? What was $\epsilon_0$]
[Area of a square 1 farad capacitor?] .
Electric double-layer capacitors (EDLC or 'supercapacitors') can have capacitances up to several thousand F.
Potential energy of a charge in an electric potential: $$U=qV$$
$$dW=V\,dq = \frac{q}{C}dq$$
$$\Rightarrow U = \frac{Q^2}{2C} = \frac{1}{2}CV^2$$
In terms of $E$-field: $$U=\frac{Q^2}{2C}=\frac{(E\epsilon_0A)^2}{2\epsilon_0 A/d}$$
$\Rightarrow$ Energy density in terms of the electric field is $$u=\frac{1}{2}\epsilon_0 |E|^2.$$
[Check this for a cylindrical capacitor this week].
(In
a second kind of dielectric, the material does not consist of dipoles, but
the field induces an average dipole moment.)
The field is reduced inside of the dielectric: $$E_D = \frac{E_0}{\kappa}$$where the dielectric constant, $\kappa$, of a material is a number greater than 1.
Permittivity: $\epsilon = \kappa \epsilon_0$
[What's the effective dielectric constant of a conductor??]
[How does capacitance change with dielectric material between plates?]
[How does energy stored change?]
If we run through our energy calculations as before,
but with a dielectric material, we find that $$u=\frac{1}{2}\kappa |E^2| =
\frac{1}{2}\epsilon |E^2|$$
Dielectric breakdown occurs in air when $|E| \gt 3\times 10^6 V/m$
[Estimate the potential difference, in volts, of our VdG with respect to ground?]
$$\myv F=k\frac{qQ}{r^2}\uv r$$ | $$k=\frac{1}{4\pi \epsilon_0}=9\times 10^9\frac{N\cdot m^2}{C^2}$$ | |||
$$\oint E_\perp \,dA=\frac{q}{\epsilon_0}$$ | $$\Phi=\int_{\cal S} E_\perp dA$$ | $$U=qV$$ | ||
$$\myv E=k\frac{q}{r^2}\uv r$$ | $$\myv E=\frac{k\lambda}{2}\frac{1}{s}\uv s$$ | $$\myv E=2\pi k\sigma\frac{\myv z}{|z|}$$ | ||
$$\Delta V = -\int \myv E \cdot d \myv l$$ | $$V=k\frac{Q}{r}$$ | $$Q=CV$$ | ||
$$u=\frac{1}{2}\epsilon_0 |E^2|$$ | $$e=1.6\times 10^{-19}\,\text{C}$$ | $$m_e=9.109\times 10^{-31}\,\text{kg}$$ |