A bar of iron is 10 cm at `20^(@)C`. At `19^(@)C` it will be (`alpha` of iron `=11xx10^-6//^(@)C`)

Options

A. `11xx10^(-6)` cm longer
B. `11xx10^(-6)` cm shorter
C. `11xx10^(-5)` cm shorte
D. `11xx10^(-5)` cm longer

Explanation

To solve the problem of finding the length of a bar of iron at 19°C, we will use the formula for linear thermal expansion. The formula is given by:

\[ \Delta L = L_0 \cdot \alpha \cdot \Delta \theta \]

Where:
- \(\Delta L\) = change in length
- \(L_0\) = original length (at 20°C)
- \(\alpha\) = coefficient of linear expansion
- \(\Delta \theta\) = change in temperature

### Step 1: Identify the given values
- Original length \(L_0 = 10 \, \text{cm}\)
- Coefficient of linear expansion \(\alpha = 11 \times 10^{-6} \, \text{°C}^{-1}\)
- Initial temperature \(T_1 = 20 \, \text{°C}\)
- Final temperature \(T_2 = 19 \, \text{°C}\)

### Step 2: Calculate the change in temperature
\[ \Delta \theta = T_2 - T_1 = 19 \, \text{°C} - 20 \, \text{°C} = -1 \, \text{°C} \]

### Step 3: Substitute the values into the thermal expansion formula
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta \theta \]
Substituting the known values:
\[ \Delta L = 10 \, \text{cm} \cdot (11 \times 10^{-6} \, \text{°C}^{-1}) \cdot (-1 \, \text{°C}) \]

### Step 4: Calculate \(\Delta L\)
\[ \Delta L = 10 \, \text{cm} \cdot 11 \times 10^{-6} \cdot (-1) = -11 \times 10^{-5} \, \text{cm} \]

### Step 5: Determine the new length of the bar
Since the change in length is negative, it indicates that the length has decreased. Therefore, the new length \(L\) at 19°C is:
\[ L = L_0 + \Delta L = 10 \, \text{cm} - 11 \times 10^{-5} \, \text{cm} \]
\[ L = 10 \, \text{cm} - 0.0011 \, \text{cm} = 9.9989 \, \text{cm} \]

### Final Answer
The length of the bar of iron at 19°C is approximately \(9.9989 \, \text{cm}\).

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