The ratio of the lengths of two wires A and B of same material is `1 :2` and the ratio of their diameters is `2:1.` They are stretched by the same force, then the ratio of increase in length will be

Options

A. `2:1`
B. `1:4`
C. `1:8`
D. `8:1`

Answer

To solve the problem, we need to find the ratio of the increase in length of two wires A and B when they are stretched by the same force. We will use the formula for the extension of a wire, which is given by:

\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]

Where:
- \(\Delta L\) = increase in length
- \(F\) = force applied
- \(L\) = original length of the wire
- \(A\) = cross-sectional area of the wire
- \(Y\) = Young's modulus of the material

### Step 1: Define the lengths and diameters of the wires
Let:
- Length of wire A = \(L_A = L\)
- Length of wire B = \(L_B = 2L\) (since the ratio of lengths is \(1:2\))

The diameters are given as:
- Diameter of wire A = \(d_A = 2d\)
- Diameter of wire B = \(d_B = d\) (since the ratio of diameters is \(2:1\))

### Step 2: Calculate the cross-sectional areas
The cross-sectional area \(A\) of a wire is given by the formula:

\[ A = \frac{\pi d^2}{4} \]

For wire A:
\[ A_A = \frac{\pi (2d)^2}{4} = \frac{\pi (4d^2)}{4} = \pi d^2 \]

For wire B:
\[ A_B = \frac{\pi (d)^2}{4} = \frac{\pi d^2}{4} \]

### Step 3: Calculate the increase in length for both wires
Using the extension formula:

For wire A:
\[ \Delta L_A = \frac{F \cdot L_A}{A_A \cdot Y} = \frac{F \cdot L}{\pi d^2 \cdot Y} \]

For wire B:
\[ \Delta L_B = \frac{F \cdot L_B}{A_B \cdot Y} = \frac{F \cdot (2L)}{\frac{\pi d^2}{4} \cdot Y} = \frac{F \cdot 2L \cdot 4}{\pi d^2 \cdot Y} = \frac{8F \cdot L}{\pi d^2 \cdot Y} \]

### Step 4: Find the ratio of increase in lengths
Now we can find the ratio of the increase in lengths:

\[ \frac{\Delta L_A}{\Delta L_B} = \frac{\frac{F \cdot L}{\pi d^2 \cdot Y}}{\frac{8F \cdot L}{\pi d^2 \cdot Y}} = \frac{1}{8} \]

### Final Answer
The ratio of the increase in length of wire A to wire B is:

\[ \Delta L_A : \Delta L_B = 1 : 8 \]