Current Division by Parallel Resistances

When a total current I_P is passed through parallel connected resistances R₁ and R₂, the voltage V_P which appears across the parallel circuit is:

V_P = I_PR_P = I_PR₁R₂ / (R₁ + R₂)

The currents I₁ and I₂ which pass through the respective resistances R₁ and R₂ are:

I₁ = V_P / R₁ = I_PR_P / R₁ = I_PR₂ / (R₁ + R₂)

I₂ = V_P / R₂ = I_PR_P / R₂ = I_PR₁ / (R₁ + R₂)

In general terms, for resistances R₁, R₂, R₃, ... (with conductances G₁, G₂, G₃, ...) connected in parallel:

V_P = I_PR_P = I_P / G_P = I_P / (G₁ + G₂ + G₃ + ...)

I_n = V_P / R_n = V_PG_n = I_PG_n / G_P = I_PG_n / (G₁ + G₂ + G₃ + ...)

where G_n = 1 / R_n

Note that the highest current passes through the highest conductance (with the lowest resistance).