Two unifrom brass rod A and B of length l and 2l and radii 2r respectively are heated to the same temperature. The ratio of the increase in the volume of A to that of B is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `1:4`

Explanation

To solve the problem of finding the ratio of the increase in the volume of two brass rods A and B when they are heated to the same temperature, we can follow these steps:

### Step 1: Determine the initial volumes of the rods
1. **Rod A** has a length \( l \) and radius \( 2r \).
- The volume \( V_A \) of rod A can be calculated using the formula for the volume of a cylinder:
\[ V_A = \pi (2r)^2 l = \pi \cdot 4r^2 \cdot l = 4\pi r^2 l \]

2. **Rod B** has a length \( 2l \) and radius \( r \).
- The volume \( V_B \) of rod B is:
\[ V_B = \pi r^2 (2l) = 2\pi r^2 l \]

### Step 2: Calculate the change in volume due to thermal expansion
The formula for the change in volume due to thermal expansion is given by:
\[ \Delta V = V \cdot \gamma \cdot \Delta T \]
where \( \gamma \) is the coefficient of volume expansion and \( \Delta T \) is the change in temperature.

1. **Change in volume for rod A**:
\[ \Delta V_A = V_A \cdot \gamma \cdot \Delta T = (4\pi r^2 l) \cdot \gamma \cdot \Delta T \]

2. **Change in volume for rod B**:
\[ \Delta V_B = V_B \cdot \gamma \cdot \Delta T = (2\pi r^2 l) \cdot \gamma \cdot \Delta T \]

### Step 3: Find the ratio of the increases in volume
Now, we can find the ratio of the increase in volume of rod A to that of rod B:
\[ \frac{\Delta V_A}{\Delta V_B} = \frac{(4\pi r^2 l) \cdot \gamma \cdot \Delta T}{(2\pi r^2 l) \cdot \gamma \cdot \Delta T} \]

### Step 4: Simplify the ratio
Notice that \( \pi \), \( r^2 \), \( l \), \( \gamma \), and \( \Delta T \) cancel out:
\[ \frac{\Delta V_A}{\Delta V_B} = \frac{4}{2} = 2 \]

### Conclusion
The ratio of the increase in the volume of rod A to that of rod B is:
\[ \frac{\Delta V_A}{\Delta V_B} = 2:1 \]