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Circuits

Circuits are typically considered element by element, where an element is a single part like a battery or a resistor. Solving circuit problems often involves setting up some kind of equation or system of equations.

Symbols in circuit analysis #symbols

$$ \begin{aligned}&{\text{Charge}}&& Q \\&{\text{Current}}&& I=\frac{\Delta Q}{\Delta t} \\&{\text{Potential difference}}&& V \\&{\text{Battery emf (in volts)}}&& \mathcal{E} \\&{\text{Resistance}}&& R \\&{\text{Capacitance}}&& C \end{aligned} $$

Warning: Voltage differences in circuits #V

For circuits only: The variable V (called "Voltage") is used to represent the electric potential difference across a single circuit element.

Resistors

The resistance of a resistor is determined by its geometry (A and L) and resistivity (𝜌):

$$ R=\frac{\rho L}{A} $$

The potential difference across a resistor pushes current through the resistor:

$$ V=IR $$

Power dissipated by a resistor:

$$ \begin{aligned}P&=IV \\ &=I^2 R \\ &=\frac{V^2}{R} \end{aligned} $$

Equivalent resistance...

$$ \begin{aligned}&{\text{In series:}}&& R_{eq}=R_1+R_2+\cdots \\&{\text{In parallel:}}&& \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots \end{aligned} $$

Capacitors

The capacitance of a parallel plate capacitor is determined by its geometry (A and d) and the dielectric constant of the material between the plates (𝜅):

$$ C=\kappa \frac{\varepsilon_0 A}{d} $$

Electric field between the capacitor plates (in vacuum):

$$ |\vec{E}|=\frac{Q}{\varepsilon_0 A} $$

The potential difference across a capacitor stores charge in that capacitor:

$$ V=CQ $$

Potential energy stored in a capacitor:

$$ \begin{aligned}U&=\frac{1}{2}QV \\ &=\frac{1}{2}CV^2 \\ &=\frac{1}{2} \frac{Q^2}{C} \end{aligned} $$

Equivalent capacitance...

$$ \begin{aligned}&{\text{In series:}}&& \frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots \\&{\text{In parallel:}}&& C_{eq}=C_1+C_2+\cdots \end{aligned} $$

Constants for capacitors

Permittivity of free space:
ε₀=8.85×10^-12 C²/Nm²

Dielectric constant of vacuum
𝜅₀=1

Circuit Analysis Rules

Kirchhoff Loop Rule: If you start at any point in a circuit and follow a loop in the circuit, the combined changes in potential across circuit elements in the loop will add up to zero.

$$ \sum V=0 $$

Kirchhoff Junction Rule: All current that flows into a junction must flow out of that junction (current is conserved)

$$ \sum I_{in} = \sum I_{out} $$

RC circuits

RC time constant

$$ \tau=RC $$

Charge, voltage, and current in an RC circuit that is charging or discharging follow exponential patterns:

Exponential functions A(t) that describe RC circuit charging or discharging, generalized for some quantity A

$$ \begin{aligned}&{\text{A starts at zero and increases:}}&& A(t)=A_\infty (1-e^{t / \tau}) \\&{\text{A starts at maximum and decays:}}&& A(t)=A_0 e^{t / \tau} \end{aligned} $$