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The current divider rule states I₂ = I_total(R₁/(R₁+R₂)) for parallel resistors, showing current distributes inversely with resistance. The rule applies when resistors share identical voltage and no external current injection occurs, revealing that lower-resistance branches carry higher currents.
You already know from Ohm's law that current through a resistor equals voltage divided by resistance: I = V/R. And from your study of resistive network combinations, you know that parallel resistors share the same voltage across their terminals. The current divider rule follows directly from combining these two facts: if the voltage across both parallel branches is identical, then the branch with lower resistance must carry more current, because I = V/R and R is smaller in that branch.
To derive the rule formally, consider two resistors R₁ and R₂ connected in parallel, with total current I_total entering the node. Since both resistors see the same terminal voltage V, the current through each is I₁ = V/R₁ and I₂ = V/R₂. The equivalent resistance of the combination is R_eq = R₁R₂/(R₁+R₂), so V = I_total × R_eq. Substituting into the expression for I₂: I₂ = V/R₂ = I_total × R_eq / R₂ = I_total × R₁/(R₁+R₂). This is the current divider formula: the fraction of total current in any branch equals the opposite resistance divided by the sum of resistances. Notice the inversion — R₁ appears in the numerator for I₂, not R₂. Current splits inversely with resistance.
This inverse relationship has a useful physical interpretation: resistance is a measure of how difficult a path is for current to flow. Given a choice between an easy path (small R) and a hard path (large R), current preferentially takes the easier route. If one branch has resistance ten times larger than the other, it carries one-tenth of the current. In the extreme, if one branch is a short circuit (R = 0), it carries all the current and the parallel branch carries none. If one branch is an open circuit (R → ∞), it carries no current and all current flows through the other branch.
The current divider generalizes beyond two branches. For N parallel resistors, the current through branch k is I_k = I_total × (G_k / G_total), where G_k = 1/R_k is the conductance of branch k and G_total is the sum of all conductances. This form shows that when working with parallel circuits, conductances add simply just as resistances add simply in series — the two formulations are duals of each other. Recognizing this duality between series (voltage divider, resistances sum) and parallel (current divider, conductances sum) circuits is a conceptual shortcut you will use throughout circuit analysis.