Resistances in Parallel

Same voltage across each resistor; currents add. For two resistors: R_eq = R₁R₂/(R₁+R₂). R_eq is always less than the smallest individual resistance.

Derivation

In parallel, the same voltage $V$ appears across each resistor. Individual currents:

$$I_1 = \frac{V}{R_1},\quad I_2 = \frac{V}{R_2},\quad \ldots,\quad I_n = \frac{V}{R_n}$$

Total current (KCL at junction):

$$I = I_1 + I_2 + \cdots + I_n = V\left(\frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}\right)$$ $$\boxed{\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}}$$

Two-resistor formula

$$R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$$

Key property

$R_{eq} < R_{min}$ — always less than the smallest resistor. Parallel connection is like increasing the cross-sectional area of a conductor.

Current divider

For two parallel resistors, current splits inversely with resistance:

$$\frac{I_1}{I_2} = \frac{R_2}{R_1}$$

The smaller resistor carries the larger current.