The volume of a metal sphere increases by `0.24%` when its temperature is raised by `40^(@)C` . The coefficient of linear expansion of the metal is .......... `.^(@)C`

Options

A. `2 xx 10^(5).^(@)C^(-1)`
B. `6 xx 10^(-5).^(@)C^(-1)`
C. `18 xx 10^(-5).^(@)C^(-1)`
D. `1.2 xx 10^(-5).^(@)C^(-1)`

Explanation

To find the coefficient of linear expansion (α) of the metal sphere, we can follow these steps:

### Step 1: Understand the relationship between volume expansion and linear expansion
The volume expansion (γ) is related to the linear expansion (α) by the formula:
\[ \gamma = 3\alpha \]
This means that the change in volume is three times the change in linear dimensions.

### Step 2: Write the formula for the change in volume
The change in volume can be expressed as:
\[ \frac{\Delta V}{V} = \gamma \Delta \theta \]
Where:
- \(\Delta V\) is the change in volume,
- \(V\) is the original volume,
- \(\Delta \theta\) is the change in temperature.

### Step 3: Substitute the known values into the equation
We know that the volume increases by \(0.24\%\), which can be expressed as:
\[ \frac{\Delta V}{V} = \frac{0.24}{100} = 0.0024 \]
The change in temperature is given as:
\[ \Delta \theta = 40^\circ C \]

### Step 4: Substitute into the volume expansion formula
Substituting the values into the volume expansion formula gives:
\[ 0.0024 = \gamma \times 40 \]

### Step 5: Solve for γ
Now, we can solve for γ:
\[ \gamma = \frac{0.0024}{40} = 0.00006 \]

### Step 6: Relate γ to α
Using the relationship \( \gamma = 3\alpha \):
\[ 0.00006 = 3\alpha \]

### Step 7: Solve for α
Now, we can solve for α:
\[ \alpha = \frac{0.00006}{3} = 0.00002 \]

### Step 8: Convert to scientific notation
Expressing this in scientific notation:
\[ \alpha = 2 \times 10^{-5} \, ^\circ C^{-1} \]

### Final Answer
Thus, the coefficient of linear expansion of the metal is:
\[ \alpha = 2 \times 10^{-5} \, ^\circ C^{-1} \]

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