A metallic bar is heated from `0^(@)C` to `100^(@)C`. The coefficient of linear expansion is `10^(-5)K^(-1)`. What will be the percentage increase in length

Options

A. `0.01%`
B. `0.1%`
C. `1%`
D. `10%`

Explanation

To solve the problem of finding the percentage increase in length of a metallic bar heated from \(0^{\circ}C\) to \(100^{\circ}C\) with a coefficient of linear expansion of \(10^{-5} K^{-1}\), we will follow these steps:

### Step 1: Understand the formula for linear expansion
The formula for the change in length (\(\Delta L\)) of a material due to temperature change is given by:
\[ \Delta L = \alpha \cdot L_0 \cdot \Delta T \]
where:
- \(\Delta L\) = change in length
- \(\alpha\) = coefficient of linear expansion
- \(L_0\) = initial length of the bar
- \(\Delta T\) = change in temperature

### Step 2: Calculate the change in temperature
The change in temperature (\(\Delta T\)) when heating from \(0^{\circ}C\) to \(100^{\circ}C\) is:
\[ \Delta T = 100^{\circ}C - 0^{\circ}C = 100 \, K \]

### Step 3: Substitute the known values into the formula
We will substitute the values into the formula for \(\Delta L\). However, we notice that we do not have the initial length (\(L_0\)), but we can express the percentage increase in length without needing its specific value:
\[ \Delta L = \alpha \cdot L_0 \cdot \Delta T = 10^{-5} \cdot L_0 \cdot 100 \]
\[ \Delta L = 10^{-3} \cdot L_0 \]

### Step 4: Find the final length
The final length (\(L_f\)) of the bar after heating can be expressed as:
\[ L_f = L_0 + \Delta L = L_0 + 10^{-3} \cdot L_0 = L_0(1 + 10^{-3}) \]

### Step 5: Calculate the percentage increase in length
The percentage increase in length is given by:
\[ \text{Percentage Increase} = \left(\frac{\Delta L}{L_0}\right) \times 100 \]
Substituting \(\Delta L\):
\[ \text{Percentage Increase} = \left(\frac{10^{-3} \cdot L_0}{L_0}\right) \times 100 = 10^{-3} \times 100 = 0.1\% \]

### Final Answer
The percentage increase in length of the metallic bar when heated from \(0^{\circ}C\) to \(100^{\circ}C\) is **0.1%**.

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