The Current Divider
With the same voltage, current splits in proportion to conductance
A source current I_in = 12 mA splits into two parallel branches, R1 = 4 kΩ and R2. Slide R2. The top bar is how the conductance splits into 1/R1 and 1/R2; the bottom bar is the current in the R2 branch taking that same share of I_in. The R2 portion of the two bars is always equally long. Grow R2 and 1/R2 shrinks, so that branch current drops. Set R2 so the R2-branch current reaches the target of 4 mA.
In parallel the same voltage sits across both
In parallel both branches are tied between the same two nodes, so the voltage V is identical. Writing Ohm’s law with conductance, the branch current is I = V/R = V·G, where G = 1/R says how easily current flows. Since the voltage V is the same for both, the ratio of the two currents I2/I1 is exactly the ratio of conductances G2/G1, that is the inverse ratio of resistances R1/R2. The branch with smaller resistance, larger conductance, takes more current. That single line — current in direct proportion to conductance — is the whole of the divider.
The branch current is the opposite resistor’s share
The total current is the sum of the branches, I_in = V·(G1+G2), so V = I_in/(G1+G2). The R2-branch current is I_2 = V·G2 = I_in·G2/(G1+G2). The divider ratio G2/(G1+G2) is the share R2’s branch holds of the total conductance. Turning conductance back into resistance, G2/(G1+G2) = (1/R2)/((1/R1)+(1/R2)) = R1/(R1+R2), so the branch current is proportional to the opposite resistor: I_2 = I_in·R1/(R1+R2). That the numerator is the opposite R1, not its own R2, is the signature of the current divider — in the voltage divider the numerator was its own R2.
The dual mirror of the voltage divider
The voltage divider split voltage by resistance ratio in series. The current divider is its mirror: it splits current by conductance ratio in parallel. Series and parallel, voltage and current, resistance and conductance are all flipped at once — a duality. So the formulas wear the same shape with only the letters swapped. The voltage divider had its own resistance on top; the current divider has the opposite resistance on top. With both tools in hand, in any mix of series and parallel you can read off the voltage at any node and the current in any branch in a single line.
Back to the first screen
The larger R2, the smaller 1/R2, so the R2-branch current bar grew shorter, reaching 4 mA when R2 was 8 kΩ (R1 = 4, I_in = 12). The share R2’s branch took in the top conductance bar and the length of the bottom current bar matched throughout. It is because the same voltage sits across both branches and the current splits in direct proportion to conductance. In resistance form I_2 = I_in·R1/(R1+R2), and that the opposite resistor R1 sits on top is what is flipped, mirror-like, from the voltage divider.