A metal rod of silver of length `100cm` at `0^(@)C` is heated to `100^(@)C` . It's length is increased by `0.19 cm` . Coefficient of cubical expansion of the silver rod is

Options

A. `5.7xx10^(-5)//C`
B. `0.63xx10^(-5)//C`
C. `1.9xx10^(-5)//C`
D. `16.1xx10^(-5)//C`

Explanation

To find the coefficient of cubical expansion of the silver rod, we can follow these steps:

### Step 1: Identify the given values
- Initial length of the rod, \( L = 100 \, \text{cm} \)
- Change in length, \( \Delta L = 0.19 \, \text{cm} \)
- Initial temperature, \( T_1 = 0^\circ C \)
- Final temperature, \( T_2 = 100^\circ C \)

### Step 2: Calculate the change in temperature
The change in temperature, \( \Delta T \), is given by:
\[ \Delta T = T_2 - T_1 = 100^\circ C - 0^\circ C = 100^\circ C \]

### Step 3: Use the formula for linear expansion
The formula for linear expansion is:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
where \( \alpha \) is the coefficient of linear expansion.

### Step 4: Rearrange the formula to solve for \( \alpha \)
Rearranging the formula gives:
\[ \alpha = \frac{\Delta L}{L \cdot \Delta T} \]

### Step 5: Substitute the known values into the equation
Substituting the known values:
\[ \alpha = \frac{0.19 \, \text{cm}}{100 \, \text{cm} \cdot 100^\circ C} \]

### Step 6: Calculate \( \alpha \)
Calculating \( \alpha \):
\[ \alpha = \frac{0.19}{10000} = 1.9 \times 10^{-5} \, \text{per} \, ^\circ C \]

### Step 7: Calculate the coefficient of cubical expansion
The coefficient of cubical expansion \( \gamma \) is related to the coefficient of linear expansion by:
\[ \gamma = 3\alpha \]
Substituting the value of \( \alpha \):
\[ \gamma = 3 \times 1.9 \times 10^{-5} = 5.7 \times 10^{-5} \, \text{per} \, ^\circ C \]

### Step 8: Conclusion
Thus, the coefficient of cubical expansion of the silver rod is:
\[ \gamma = 5.7 \times 10^{-5} \, \text{per} \, ^\circ C \]

### Final Answer
The correct option is:
**Option 1: \( 5.7 \times 10^{-5} \, \text{per} \, ^\circ C \)**

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