Current, Resistance, and DC Circuits

A copper wire carries no more charge when current flows than when it does not — the electrons were always there, drifting slower than honey (about 0.1 mm/s). What makes a bulb light instantly is that the push propagates near light speed, like water pressure through a full hose.

Electric current is the rate of charge flow:

$$I = \frac{dQ}{dt}, \qquad \text{unit: ampere (A)}$$

Ohm's law relates voltage, current, and resistance:

V = IR

Resistivity — $R = \rho L/A$, where $\rho$ is the material's resistivity, $L$ is length, and $A$ is cross-sectional area.

Power dissipated in a resistor:

$$P = IV = I^2R = \frac{V^2}{R}$$

Resistor combinations

Joule heating — Energy dissipated as heat: $E = Pt = I^{2}Rt$. This is why transmission lines use high voltage and low current.

V = 12.0   # volts (battery)
R1, R2 = 100, 200  # ohms in series
I = V / (R1 + R2)
P = I**2 * (R1 + R2)
print(f"Current = {I:.4f} A, Power = {P:.4f} W")

Key insight: Resistors in series share the same current; resistors in parallel share the same voltage. This determines which combination rule to use.

Common pitfall: Current is not "used up" around a circuit. The same current that leaves the battery returns to it; what gets spent is energy per charge (voltage). Series bulbs share voltage, not diminishing current.