The Voltage Divider
With the same current, voltage splits in proportion to resistance
Wire R1 = 4 kΩ and R2 in series across V_in = 12 V, and take the output across R2. Slide R2. The top bar is how the resistance splits into R1 and R2; the bottom bar is the output voltage taking that same share of V_in. The R2 portion of the two bars is always equally long. Set R2 so the output reaches the target of 8 V.
The same current passes through both resistors
In series there is one unbroken path, so the same current I = V_in/(R1+R2) passes through both resistors. By Ohm’s law the voltage across each is V1 = I·R1 and V2 = I·R2. Since the current I is identical for both, the ratio of the two voltages V2/V1 is exactly the ratio of the resistances R2/R1. The larger resistance holds the larger voltage. That single line — voltage in direct proportion to resistance — is the whole of the divider.
The output is R2’s share
Take the output across R2 and V_out = V2 = I·R2; substituting I = V_in/(R1+R2) gives V_out = V_in·R2/(R1+R2). Here the divider ratio R2/(R1+R2) is exactly the fraction of the total resistance that R2 occupies. When R2 is far larger than R1 the ratio nears 1 and the output reaches V_in; when R2 is small the ratio nears 0 and the output nearly vanishes. Growing R2 is growing R2’s share of the voltage.
A tool to read, and the mirror of current division
The voltage divider is a staple tool of series–parallel analysis: making a reference voltage, or scaling a large voltage down to a safe size to read. It holds as written only when the load current drawn at the output is small; a heavy load disturbs the ratio, so exams take the no-load divider as the baseline. Within the same analysis, current division — where parallel branches split the current in proportion to conductance — is the mirror of this. Series splits voltage by resistance ratio; parallel splits current by conductance ratio.
Back to the first screen
The larger R2, the longer the output bar grew, reaching 8 V when R2 was 8 kΩ (R1 = 4, V_in = 12). The share R2 took in the top resistance bar and the length of the bottom output bar matched throughout. It is because the same current passes through both resistors and the voltage splits in direct proportion to resistance. The divider ratio R2/(R1+R2) — the share R2 holds of the total resistance — sets the whole of the output voltage.