Resistances in Series

Same current through each resistor; voltages add. Equivalent resistance is the sum. R_eq is always greater than the largest individual resistance.

Derivation

In series, the same current $I$ flows through each resistor (no branch points). Voltage drops:

$$V_1 = IR_1,\quad V_2 = IR_2,\quad \ldots,\quad V_n = IR_n$$

Total voltage:

$$V = V_1 + V_2 + \cdots + V_n = I(R_1 + R_2 + \cdots + R_n)$$

Equivalent resistance $R_{eq} = V/I$:

$$\boxed{R_{eq} = R_1 + R_2 + \cdots + R_n}$$

Key property

$R_{eq} > R_{max}$ — always greater than the largest individual resistor. Each resistor adds to the total opposition.

Physical picture

Series connection is like extending the length of a conductor — $R \propto L$, so total resistance adds directly.

Current divider (contrast)

In series, voltage divides: $V_i/V = R_i/R_{eq}$. The largest resistor takes the largest share of the voltage.