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 finding the expansion ratio \( E_h : E_s \) for a hollow rod and a solid rod made of the same material, we can follow these steps:

### Step 1: Understand the Problem
We have two rods: one is hollow and the other is solid. Both rods have the same length (20 cm) and the same radius (2 cm). They are made of the same material and are subjected to the same temperature increase of \( 50^\circ C \).

### Step 2: Identify the Relevant Formula
The change in length (\( \Delta L \)) of a rod due to thermal expansion can be expressed using the formula:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
where:
- \( L \) = original length of the rod,
- \( \alpha \) = coefficient of linear expansion of the material,
- \( \Delta T \) = change in temperature.

### Step 3: Apply the Formula to Both Rods
For both rods (hollow and solid), since they have the same length, material, and temperature change, we can denote:
- \( \Delta L_h \) for the hollow rod,
- \( \Delta L_s \) for the solid rod.

Thus, we have:
\[ \Delta L_h = L \cdot \alpha \cdot \Delta T \]
\[ \Delta L_s = L \cdot \alpha \cdot \Delta T \]

### Step 4: Compare the Expansions
Since both rods have the same length \( L \), the same coefficient of linear expansion \( \alpha \), and the same temperature change \( \Delta T \), it follows that:
\[ \Delta L_h = \Delta L_s \]

### Step 5: Determine the Expansion Ratio
The expansion ratio \( E_h : E_s \) is given by:
\[ E_h : E_s = \Delta L_h : \Delta L_s = 1 : 1 \]

### Conclusion
Thus, the expansion ratio \( E_h : E_s \) is \( 1 : 1 \).