Electric Circuits
Core concepts
Circuits questions test Ohm's law ($V=IR$), power ($P=IV=I^2R=V^2/R$), and rules for series and parallel combinations. In series, current is the same and resistances add; in parallel, voltage is the same and $1/R_{eq}=\sum 1/R_i$. Kirchhoff's junction rule (charge conservation) and loop rule (energy conservation) solve multi-loop circuits. For capacitors, $Q=CV$, energy stored $U=\tfrac12CV^2$, and combinations reverse resistors: series capacitors add as $1/C_{eq}=\sum 1/C_i$, parallel capacitors add directly. RC circuits charge and discharge with time constant $\tau=RC$. Internal resistance of a battery reduces terminal voltage: $V_{term}=\varepsilon - Ir$.
Traps to watch
- Mixing up the series/parallel rules for capacitors versus resistors.
- Assuming current is "used up" as it passes through resistors; it is not-charge is conserved.
- Forgetting that an ideal ammeter has ~zero resistance (placed in series) and an ideal voltmeter has ~infinite resistance (placed in parallel).
- In parallel branches, the branch with smaller resistance carries more current.
- Power dissipated depends on which quantity is fixed: at constant $V$ use $V^2/R$; at constant $I$ use $I^2R$.
Worked example
A $12,\mathrm{V}$ battery with internal resistance $1,\Omega$ drives a $5,\Omega$ resistor. Total resistance is $6,\Omega$, so $I=2,\mathrm{A}$. Terminal voltage is $12-(2)(1)=10,\mathrm{V}$, and power in the external resistor is $I^2R=(4)(5)=20,\mathrm{W}$.
Rule of thumb: reduce the network to one equivalent resistance, find the total current, then work back outward.