Two uniform brass rods A and B of lengths l and 2l and radii 2r and r respectively are heated to the same temperature. The ratio of the increase in the length of A to that of B is :

Options

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

Explanation

To solve the problem of finding the ratio of the increase in length of two brass rods A and B when heated to the same temperature, we can use the formula for linear expansion. The formula for the change in length (ΔL) due to thermal expansion is given by:

\[ \Delta L = \alpha L \Delta T \]

where:
- \(\Delta L\) = change in length
- \(\alpha\) = coefficient of linear expansion
- \(L\) = original length of the rod
- \(\Delta T\) = change in temperature

### Step 1: Identify the parameters for both rods

For rod A:
- Length, \(L_A = l\)
- Coefficient of linear expansion, \(\alpha_A = \alpha\) (since both rods are made of brass, they have the same coefficient)
- Change in temperature, \(\Delta T\) (same for both rods)

For rod B:
- Length, \(L_B = 2l\)
- Coefficient of linear expansion, \(\alpha_B = \alpha\) (same as rod A)
- Change in temperature, \(\Delta T\) (same for both rods)

### Step 2: Calculate the change in length for both rods

For rod A:
\[ \Delta L_A = \alpha L_A \Delta T = \alpha l \Delta T \]

For rod B:
\[ \Delta L_B = \alpha L_B \Delta T = \alpha (2l) \Delta T = 2\alpha l \Delta T \]

### Step 3: Find the ratio of the increase in length

Now, we can find the ratio of the increase in length of rod A to that of rod B:

\[ \frac{\Delta L_A}{\Delta L_B} = \frac{\alpha l \Delta T}{2\alpha l \Delta T} \]

### Step 4: Simplify the ratio

The \(\alpha\), \(l\), and \(\Delta T\) terms cancel out:

\[ \frac{\Delta L_A}{\Delta L_B} = \frac{1}{2} \]

### Conclusion

Thus, the ratio of the increase in length of rod A to that of rod B is:

\[ \frac{\Delta L_A}{\Delta L_B} = 1 : 2 \]