Electric Circuits
Core concepts
Ohm's law (V = IR) applies to ohmic resistors. Kirchhoff's rules govern any network: the junction rule (charge conservation) and the loop rule (energy conservation, (\sum \Delta V = 0)). Resistors in series add; in parallel the reciprocals add. Power dissipated is (P = I^2R = V^2/R).
For RC circuits, charging follows (q(t) = C\varepsilon(1 - e^{-t/RC})) and current decays as (I = (\varepsilon/R)e^{-t/RC}), with time constant (\tau = RC). At (t = 0) an uncharged capacitor acts like a wire; after a long time it acts like an open circuit. Ammeters are placed in series with low resistance; voltmeters in parallel with high resistance.
Traps
- Applying the loop rule without consistent sign conventions; pick a direction and stick to it.
- Forgetting that a battery's internal resistance reduces terminal voltage: (V_{\text{term}} = \varepsilon - Ir).
- Assuming a capacitor blocks current instantly; it blocks steady current but passes transient current.
- Mixing up series/parallel rules for resistors versus capacitors.
- Ignoring that ideal voltmeters draw no current, while real ones perturb the circuit.
Worked example
A (9,\text{V}) battery with (r = 1,\Omega) drives a (2,\Omega) resistor. Total resistance (3,\Omega), so (I = 3,\text{A}). Terminal voltage is (9 - 3(1) = 6,\text{V}). Power in the external resistor: (I^2R = 18,\text{W}). Rule of thumb: always fold internal resistance into the total before computing current, then subtract (Ir) to get terminal voltage.