A liquid of coefficient of viscosity `eta` is flowing steadily through a capillary tube of radius r and length I. If V is volume of liquid flowing per sec. the pressure difference P at the end of tube is given by

Options

A. `P =(8piIV)/(eta r^4)`
B. `P = (8eta r^4 I)/(piV)`
C. `P = (8etaIV)/(pir^4)`
D. `P =(8eta r^4 V)/(pi I)`

Explanation

To derive the pressure difference \( P \) at the end of a capillary tube through which a liquid of viscosity \( \eta \) is flowing, we can use the principles of fluid dynamics, specifically Poiseuille's law. Here’s the step-by-step solution:

### Step 1: Understand the parameters involved
We have:
- Coefficient of viscosity \( \eta \)
- Radius of the capillary tube \( r \)
- Length of the capillary tube \( l \)
- Volume of liquid flowing per second \( V \)
- Pressure difference \( P \)

### Step 2: Recall Poiseuille's Law
According to Poiseuille's law, the volume flow rate \( Q \) (which is the volume \( V \) flowing per second) through a capillary tube is given by:

\[ Q = \frac{\pi r^4 (P)}{8 \eta l} \]

Where:
- \( P \) is the pressure difference across the length \( l \) of the tube.
- \( \eta \) is the viscosity of the liquid.
- \( r \) is the radius of the tube.
- \( l \) is the length of the tube.

### Step 3: Rearranging the equation for pressure difference \( P \)
From the equation above, we can express the pressure difference \( P \) as:

\[ P = \frac{8 \eta l Q}{\pi r^4} \]

Since \( Q = V \) (the volume of liquid flowing per second), we can substitute \( Q \) with \( V \):

\[ P = \frac{8 \eta l V}{\pi r^4} \]

### Step 4: Final expression for pressure difference
Thus, the pressure difference \( P \) at the end of the tube is given by:

\[ P = \frac{8 \eta l V}{\pi r^4} \]

### Summary
The final expression for the pressure difference \( P \) at the end of the capillary tube is:

\[ P = \frac{8 \eta l V}{\pi r^4} \]