In Poiseuille's law, how does resistance relate to radius?

• A. R is proportional to r^4
• B. R is proportional to 1/r^4
• C. R is proportional to length
• D. R is independent of radius

Answer: (B)

In Poiseuille's law, the resistance to flow through a cylindrical vessel is determined by how radius influences flow. The law states that the flow rate Q is proportional to r^4, specifically Q = (π ΔP r^4)/(8 μ L). If you define resistance as R = ΔP / Q, you get R = (8 μ L)/(π r^4). This shows resistance is inversely proportional to the fourth power of the radius.

So, as the radius grows, resistance drops very quickly: doubling the radius reduces resistance by a factor of 16; halving the radius increases resistance by 16 times. Viscosity and length also affect resistance (R ∝ μ L / r^4), but for a given fluid and vessel length, the radius is the dominant geometric factor. This explains why small changes in vessel radius markedly alter flow.