Circuits

Ohm's Law and Resistance

Current is charge flow per time, driven by voltage against resistance: $V = IR$ Resistance scales with geometry and material: $R = \rho L/A$ — longer wires resist more, thicker wires less. Kirchhoff's junction rule conserves charge (currents into a node equal currents out); the loop rule conserves energy (voltage drops around a loop sum to the EMF).

Series and Parallel

Series resistors share one current and add directly: $R_{eq} = R_1 + R_2 + \cdots$. Parallel resistors share the full voltage and combine reciprocally: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$ so the equivalent is smaller than any branch. Power dissipation is $P = IV = I^2R = \frac{V^2}{R}$

Capacitors

A capacitor stores charge and energy: $C = Q/V$, with $U = \tfrac{1}{2}CV^2$. Parallel-plate capacitance is $C = \kappa\epsilon_0 A/d$, and a dielectric raises it. Capacitors add oppositely to resistors: directly in parallel, reciprocally in series. In RC behavior, an uncharged capacitor initially acts like a wire and, once charged, like a break in the circuit.

Key Takeaways

• Series: same current, voltages divide. Parallel: same voltage, currents divide.
• Adding a parallel resistor always lowers total resistance.
• Use whichever form of $P$ matches the known quantities.
• Capacitor combination rules are the reverse of resistor rules.