PHYS 208 Lecture 8

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Electric Potential (Cont'd)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} U &= \frac{kq_1q_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{align}}

Example (Dr. Webb's Way)

Given a line segment of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2L} along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis and centered at the origin with a total charge of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} , what is the potential due to this charge along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} -axis.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) = \ldots}

Same Example (My Way)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} separated by a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} :

By Gauss' law, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)