Two rods, one hollow and the other solid, made of the same material have the same length of 20 cm and radius of 2 cm. When their temperature is increased through the same amount of `50^(@)C`, their expansion ratio Eh : Es will be :

Options

A. `1 : 8`
B. ` 1 : 4`
C. `1 : 1`
D. `1 : 2`

Explanation

To solve the problem of determining the expansion ratio \(E_h : E_s\) of a hollow rod to a solid rod made of the same material, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two rods (one hollow and one solid) made of the same material, both having the same length (20 cm) and radius (2 cm). We need to find the ratio of their expansions when the temperature is increased by \(50^\circ C\).

2. **Identify the Formula for Linear Expansion**: The linear expansion of a rod can be described by the formula:
\[ \Delta L = L_0 \alpha \Delta T \]
where:
- \(\Delta L\) = change in length
- \(L_0\) = original length of the rod
- \(\alpha\) = coefficient of linear expansion of the material
- \(\Delta T\) = change in temperature

3. **Apply the Formula to Both Rods**:
- For the hollow rod:
\[ \Delta L_h = L_0 \alpha \Delta T \]
- For the solid rod:
\[ \Delta L_s = L_0 \alpha \Delta T \]

4. **Recognize that the Lengths and Coefficients are the Same**: Since both rods are made of the same material and have the same original length and temperature change:
- \(L_0\) is the same for both rods.
- \(\alpha\) is the same for both rods.
- \(\Delta T\) is the same for both rods.

5. **Calculate the Change in Length for Both Rods**: Since all factors are the same, we can conclude:
\[ \Delta L_h = \Delta L_s \]

6. **Find the Ratio of Expansions**:
- The expansion ratio \(E_h : E_s\) can be expressed as:
\[ E_h : E_s = \Delta L_h : \Delta L_s = 1 : 1 \]

7. **Conclusion**: The expansion ratio of the hollow rod to the solid rod is:
\[ E_h : E_s = 1 : 1 \]

### Final Answer:
The expansion ratio \(E_h : E_s\) is \(1 : 1\).