The rate of flow of a liquid through a capillary tube of radius $r$ and length $l$ under a pressure difference $P$ is proportional to
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Remembering the key components of Poiseuille's Law—pressure difference, radius raised to the fourth power, and length in the denominator—is crucial for solving such problems.
Step 1: Concept
The rate of flow (or volumetric flow rate) of a liquid through a capillary tube can be described by Poiseuille's Law. This law states that the flow rate is directly proportional to the pressure difference across the ends of the tube, inversely proportional to its length, and directly proportional to the fourth power of the radius of the tube.
Step 2: Meaning
In mathematical terms, if \( Q \) represents the volumetric flow rate, then according to Poiseuille's Law:
\[Q = \frac{\pi P r^4}{8 \eta l}\]
where:
\( Q \) is the flow rate,
\( P \) is the pressure difference across the tube,
\( r \) is the radius of the capillary tube,
\( l \) is the length of the tube,
\( \eta \) is the dynamic viscosity of the liquid.
Step 3: Analysis
To determine which option correctly represents the relationship, we need to focus on the terms involving \( r \) and \( l \). From Poiseuille's Law:
\[Q \propto P r^4 / l\]
This proportionality indicates that the flow rate is directly proportional to the fourth power of the radius (\( r^4 \)) and inversely proportional to the length (\( l \)). Therefore, the correct expression for the rate of flow must include \( r^4 / l \).
Option A: \( r^4 / l \) matches this relationship.
Options B, C, and D do not match because they either involve a different power of \( r \) or do not account for the fourth power dependency on radius as required by Poiseuille's Law.
Step 4: Conclusion
The rate of flow is directly proportional to \( r^4 / l \).
Final Answer: (A)
