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.

$C={\frac {Q}{V_{ab}}}$

Example: Spherical capacitor

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

$V=-\int _{r_{b}}^{r_{a}}{\frac {q}{4\pi \epsilon _{0}r^{2}}}\mathrm {d} r={\frac {q}{4\pi \epsilon _{0}}}\,\left({\frac {1}{r_{a}}}-{\frac {1}{r_{b}}}\right)$

$C={\frac {4\pi \epsilon _{0}}{\left({\frac {1}{r_{a}}}-{\frac {1}{r_{b}}}\right)}}$

Capacitors in Circuits

Multiple capacitors in a circuit can be replaced with a single capacitor with equivalent capacitance $C_{eq}$.

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 $Q=CV$ or $V=Q/C$ is held constant.

Series

Charge $Q$ remains constant through serially connected capacitors (and thus on the equivalent capacitor)

Potential $V$ varies from capacitor to capacitor depending on capacitance, but the potential across each capacitor will add to the total voltage applied.

${\frac {1}{C_{eq}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\dots$

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 $V$ remains constant through each capacitor.

$C_{eq}=C_{1}+C_{2}+\dots$

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.

${\begin{aligned}Q_{total}&=Q_{1}+Q_{eq}\\V&={\frac {Q_{1}}{C_{1}}}+{\frac {Q_{eq}}{C_{eq}}}\end{aligned}}$