Two metallic wires P and Q have same volume and are made up of same material. If their area of cross sections are in the ratio 4 : 1 and force `F_1` is applied to P, an extension of `Delta l` is produced. The force which is required to produce same extension in Q is `F_2`.
The value of `frac{F_1}{F_2}` is_______ .

Answer

To solve the problem, we need to find the ratio \( \frac{F_1}{F_2} \) where \( F_1 \) is the force applied to wire P and \( F_2 \) is the force applied to wire Q. Both wires have the same volume and are made of the same material, but their areas of cross-section are in the ratio 4:1.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Let the area of cross-section of wire P be \( A_1 \) and that of wire Q be \( A_2 \).
- Given the ratio of the areas:
\[ \frac{A_1}{A_2} = \frac{4}{1} \implies A_1 = 4A_2 \]

2. **Using the Volume Constraint**:
- Since both wires have the same volume, we can express the volume \( V \) as:
\[ V = A_1 L_1 = A_2 L_2 \]
- Substituting \( A_1 = 4A_2 \) into the volume equation:
\[ 4A_2 L_1 = A_2 L_2 \]
- Dividing both sides by \( A_2 \) (assuming \( A_2 \neq 0 \)):
\[ 4L_1 = L_2 \implies L_2 = 4L_1 \]

3. **Applying Young's Modulus**:
- Young's modulus \( \gamma \) is defined as:
\[ \gamma = \frac{F L}{A \Delta L} \]
- For wire P:
\[ \gamma = \frac{F_1 L_1}{A_1 \Delta L} \]
- For wire Q:
\[ \gamma = \frac{F_2 L_2}{A_2 \Delta L} \]

4. **Setting the Equations Equal**:
- Since both wires are made of the same material, their Young's modulus is the same:
\[ \frac{F_1 L_1}{A_1 \Delta L} = \frac{F_2 L_2}{A_2 \Delta L} \]
- Canceling \( \Delta L \) from both sides:
\[ \frac{F_1 L_1}{A_1} = \frac{F_2 L_2}{A_2} \]

5. **Substituting Known Values**:
- Substitute \( L_2 = 4L_1 \) and \( A_1 = 4A_2 \):
\[ \frac{F_1 L_1}{4A_2} = \frac{F_2 (4L_1)}{A_2} \]
- Cancel \( A_2 \) and \( L_1 \) (assuming they are not zero):
\[ \frac{F_1}{4} = 4F_2 \]

6. **Finding the Ratio**:
- Rearranging gives:
\[ F_1 = 16 F_2 \]
- Therefore, the ratio is:
\[ \frac{F_1}{F_2} = 16 \]

### Final Answer:
\[ \frac{F_1}{F_2} = 16 \]