PHYS 208 Lecture 8

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Electric Potential (Cont'd)

${\begin{aligned}U&={\frac {kq_{1}q_{2}}{r}}=qV\\V&={\frac {kq}{r}}\\E&=-{\frac {\mathrm {d} V}{\mathrm {d} r}}\\{\vec {E}}&=-{\vec {\nabla }}V=-\left\langle {\frac {\partial V}{\partial x}},\,{\frac {\partial V}{\partial y}},\,{\frac {\partial V}{\partial z}}\right\rangle \end{aligned}}$

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.

${\begin{aligned}V(x)&=\int _{-L}^{+L}{\frac {k\lambda \mathrm {d} x'}{((x-x')^{2})^{1/2}}}\\&=-k\lambda \ln(x-x'){\big |}_{-L}^{+L}\\&=k\lambda \ln \left({\frac {x+L}{x-L}}\right)\end{aligned}}$

$V(x)=\ldots$

Same Example (My Way)

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

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 20px 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, $E=\sigma /\epsilon _{0}$ inside the capacitor (where σ is the surface charge density of a plate inside the conductor)

The potential difference between the terminals must be

$\Delta V={\frac {Q}{V}}=\epsilon _{0}{\frac {A}{d}}$

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