PHYS 208 Lecture 8

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Electric Potential (Cont'd)

$\begin{array}{rl}U& =\frac{k{q}_1{q}_2}{r}=qV\\ V& =\frac{kq}{r}\\ E& =-\frac{\mathrm{d}V}{\mathrm{d}r}\\ \overrightarrow{E}& =-\overrightarrow{\mathrm{\nabla }}V=-⟨\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z}⟩\end{array}$

Example (Dr. Webb's Way)

Given a line segment of length $2L$ along the $x$-axis and centered at the origin with a total charge of $Q$, what is the potential due to this charge along the $x$-axis and the $y$-axis.

$\begin{array}{rl}V(x)& =\underset{-L}{\overset{+L}{\int }}\frac{k\lambda \mathrm{d}{x}^{\prime }}{((x-x^{\prime }{)}^2{)}^{1/2}}\\ & =-k\lambda \ln (x-x^{\prime }){|}_{-L}^{+L}\\ & =k\lambda \ln \left(\frac{x+L}{x-L}\right)\end{array}$

$V(x)=\dots$

Same Example (My Way)

$\begin{array}{rl}V(y)& =2\underset{0}{\overset{L}{\int }}\frac{k\lambda \mathrm{d}y}{\sqrt{x^2+y^2}}\cos \theta \\ & =2\underset{0}{\overset{L}{\int }}\frac{k\lambda \mathrm{d}y}{\sqrt{x^2+y^2}}\frac{y}{\sqrt{x^2+y^2}}\\ & =2\underset{0}{\overset{L}{\int }}\frac{ky\lambda }{x^2+y^2}\mathrm{d}y\end{array}$

Ch. 24: Capacitance, Dielectrics, Electric Energy Storage

capacitor (condenser)

an device that we can store charge and/or electric energy in a circuit

Two conductors of equal and opposite charge separated by an insulator or a vacuum

Represented by File:Cappacitor Symbol.svg in circuit diagrams

There will be an electric field between the terminals and a potential difference between them. This can be defined mathematically by

$C=\frac{Q}{V}$

Capacitance is measured in faradays: [C/V] = [farad] = [F]

Most capacitors nowadays are in microfarads and nanofarads

Equations

Two parallel plates with an area $A$ separated by a distance $d$:

By Gauss' law, $E=\sigma /{ϵ}_0$ inside the capacitor (where σ is the surface charge density of a plate inside the conductor)

The potential difference between the terminals must be

$\mathrm{\Delta }V=\frac{Q}{V}={ϵ}_0\frac{A}{d}$

ε₀ is a constant for a vacuum. if there is an insulating material between the plates, this constant changes (called dielectric constant)