PHYS 208 Lecture 9
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Capacitance and Dielectrics (Cont'd)
From last time, a capacitor is composed of two conductors separated by an insulator and can hold a charge.
Used to filter/smooth out voltages.
Capacitance is related to the spring constant in mechanics.
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_{ab}}}
Example: Spherical capacitor
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}}
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 = -\int_{r_b}^{r_a} \frac{q}{4\pi\epsilon_0 r^2} \mathrm{d}r = \frac{q}{4\pi\epsilon_0}\,\left(\frac{1}{r_a}-\frac{1}{r_b}\right)}
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{ 4 \pi \epsilon_0 }{ \left( \frac{1}{r_a} - \frac{1}{r_b} \right) }}
Capacitors in Circuits
Multiple capacitors in a circuit can be replaced with a single capacitor with equivalent capacitance 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_{eq}} .
There are two basic types of circuits: series and parallel. For more complicated circuits, break the configuration into groups of series and parallel, finding the equivalent capacitance of each subunit, then using that value in the equivalent capacitance of the next larger system. Repeat until the entire circuit is represented as a single capacitor.
When asked to solve for voltage and charge across each capacitor, solve for the equivalent capacitor, then break apart and solve for next smaller subunit using properties (which 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=CV} or 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=Q/C} is held constant.
Series
Charge 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} remains constant through serially connected capacitors (and thus on the equivalent capacitor)
Potential 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} varies from capacitor to capacitor depending on capacitance, but the potential across each capacitor will add to the total voltage applied.
Parallel
Charge is independent for each capacitor and depends on capacitance. Sum of individual charges will equal the total charge on the equivalent capacitor
Potential 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} remains constant through each capacitor.
Redistributed Charge
Charge applied to one capacitor in circuit, then circuit directed to other capacitors so the first acts like a battery. Therefore, voltage must remain constant between charged capacitor and equivalent of others.
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} Q_{total} &= Q_1 + Q_{eq} \\ V &= \frac{Q_1}{C_1} + \frac{Q_{eq}}{C_{eq}} \end{align}}