A wire X is half the diameter and half the length of a wire Y of similar material. The ratio of resistance of X to that of Y is

Options

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

Explanation

To find the ratio of the resistance of wire X to that of wire Y, we can follow these steps:

### Step 1: Understand the given dimensions
- Let the diameter of wire Y be \( D \) and its length be \( L \).
- Therefore, the diameter of wire X is \( \frac{D}{2} \) and its length is \( \frac{L}{2} \).

### Step 2: Calculate the cross-sectional areas
- The cross-sectional area \( A \) of a wire can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
- For wire Y:
- Radius \( r_Y = \frac{D}{2} \)
- Area \( A_Y = \pi \left(\frac{D}{2}\right)^2 = \pi \frac{D^2}{4} \)

- For wire X:
- Radius \( r_X = \frac{D}{4} \) (since the diameter is \( \frac{D}{2} \))
- Area \( A_X = \pi \left(\frac{D}{4}\right)^2 = \pi \frac{D^2}{16} \)

### Step 3: Write the resistance formulas
- The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material.

- For wire Y:
\[ R_Y = \frac{\rho L}{A_Y} = \frac{\rho L}{\pi \frac{D^2}{4}} = \frac{4\rho L}{\pi D^2} \]

- For wire X:
\[ R_X = \frac{\rho L_X}{A_X} = \frac{\rho \frac{L}{2}}{\pi \frac{D^2}{16}} = \frac{16\rho \frac{L}{2}}{\pi D^2} = \frac{8\rho L}{\pi D^2} \]

### Step 4: Calculate the ratio of the resistances
- Now, we need to find the ratio \( \frac{R_X}{R_Y} \):
\[ \frac{R_X}{R_Y} = \frac{\frac{8\rho L}{\pi D^2}}{\frac{4\rho L}{\pi D^2}} = \frac{8}{4} = 2 \]

### Conclusion
The ratio of the resistance of wire X to that of wire Y is:
\[ \frac{R_X}{R_Y} = 2:1 \]