The electrical resitance of a mercury column in a cylindrical container with half the radius of cross-section. The resistance of the mercury column is

Options

A. R
B. 2R
C. 16R
D. 5R

Explanation

To solve the problem of finding the electrical resistance of a mercury column in a cylindrical container with half the radius of cross-section, we can follow these steps:

### Step 1: Understand the formula for resistance
The resistance \( R \) of a cylindrical conductor can be calculated using the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the conductor,
- \( A \) is the cross-sectional area.

### Step 2: Determine the cross-sectional area of the original and new cylinder
For a cylinder with radius \( R \), the cross-sectional area \( A \) is given by:
\[ A = \pi R^2 \]
For a cylinder with half the radius \( \frac{R}{2} \), the new cross-sectional area \( A_1 \) is:
\[ A_1 = \pi \left(\frac{R}{2}\right)^2 = \pi \frac{R^2}{4} = \frac{A}{4} \]

### Step 3: Relate the lengths of the two columns
Assuming both columns have the same height (length), let’s denote the length of the original mercury column as \( L \). The length of the mercury column in the new container remains the same, so:
\[ L_1 = L \]
For the new cylinder with area \( A_1 \), the resistance \( R_1 \) can be expressed as:
\[ R_1 = \frac{\rho L}{A_1} \]
Substituting \( A_1 \):
\[ R_1 = \frac{\rho L}{\frac{A}{4}} = \frac{4 \rho L}{A} \]

### Step 4: Relate the new resistance to the original resistance
From the original resistance \( R \):
\[ R = \frac{\rho L}{A} \]
Thus, we can express \( R_1 \) in terms of \( R \):
\[ R_1 = 4 \cdot \frac{\rho L}{A} = 4R \]

### Step 5: Calculate the final resistance
Since the new cylinder has a radius that is half of the original, the resistance of the mercury column in the new container is:
\[ R_1 = 4R \]

### Step 6: Conclusion
The electrical resistance of the mercury column in the cylindrical container with half the radius of cross-section is \( 4R \).