Two capillary tubes of lengths in the ratio `2 : 1` and radii in the ratio `1 : 2` are connected in series. Assume the flow of the liquid through the tube is steady. Then, the ratio of pressure difference across the tubes is

Options

A. `1 : 8`
B. `1 : 16`
C. `32 : 1`
D. `1 : 1`

Explanation

To solve the problem of finding the ratio of pressure difference across two capillary tubes connected in series, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The lengths of the two capillary tubes are in the ratio \( L_1 : L_2 = 2 : 1 \).
- The radii of the two capillary tubes are in the ratio \( R_1 : R_2 = 1 : 2 \).

2. **Express Lengths and Radii**:
- Let \( L_2 = L \). Then, from the ratio, \( L_1 = 2L \).
- Let \( R_2 = R \). Then, from the ratio, \( R_1 = \frac{R}{2} \).

3. **Use the Pressure Difference Formula**:
- The pressure difference \( \Delta P \) across a capillary tube is given by:
\[ \Delta P \propto \frac{R^4}{L} \]
- Therefore, for the first tube (length \( L_1 \) and radius \( R_1 \)):
\[ \Delta P_1 \propto \frac{R_1^4}{L_1} \]
- For the second tube (length \( L_2 \) and radius \( R_2 \)):
\[ \Delta P_2 \propto \frac{R_2^4}{L_2} \]

4. **Calculate the Pressure Differences**:
- Substitute \( R_1 \) and \( L_1 \):
\[ \Delta P_1 \propto \frac{\left(\frac{R}{2}\right)^4}{2L} = \frac{\frac{R^4}{16}}{2L} = \frac{R^4}{32L} \]
- Substitute \( R_2 \) and \( L_2 \):
\[ \Delta P_2 \propto \frac{R^4}{L} \]

5. **Find the Ratio of Pressure Differences**:
- Now, we can find the ratio \( \frac{\Delta P_1}{\Delta P_2} \):
\[ \frac{\Delta P_1}{\Delta P_2} = \frac{\frac{R^4}{32L}}{\frac{R^4}{L}} = \frac{1}{32} \]

6. **Reciprocal for the Final Ratio**:
- The ratio \( \frac{\Delta P_2}{\Delta P_1} \) is the reciprocal:
\[ \frac{\Delta P_2}{\Delta P_1} = 32 \]
- Thus, the final ratio of pressure differences across the tubes is:
\[ \Delta P_2 : \Delta P_1 = 32 : 1 \]

### Final Answer:
The ratio of pressure difference across the tubes is \( 32 : 1 \).

---