Two wires made of same material have lengths in the ratio `1:2` and their volumes in the same ratio. The ratio of their resistances is

Options

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

Explanation

To find the ratio of the resistances of two wires made of the same material, with lengths and volumes in the ratio of 1:2, we can follow these steps:

### Step 1: Define the lengths and volumes of the wires
Let the length of the first wire (L1) be \( L \) and the length of the second wire (L2) be \( 2L \). Since the volumes of the wires are also in the ratio of 1:2, we can denote the volume of the first wire (V1) as \( V \) and the volume of the second wire (V2) as \( 2V \).

### Step 2: Express the volume in terms of length and cross-sectional area
The volume (V) of a wire can be expressed as:
\[ V = L \times A \]
where \( A \) is the cross-sectional area of the wire.

For the first wire:
\[ V1 = L1 \times A1 = L \times A1 \]
For the second wire:
\[ V2 = L2 \times A2 = 2L \times A2 \]

### Step 3: Set the volumes equal to each other
Since the volumes are in the ratio of 1:2, we can write:
\[ L \times A1 = 2L \times A2 \]

### Step 4: Simplify the equation
Dividing both sides by \( L \) (assuming \( L \neq 0 \)):
\[ A1 = 2A2 \]
This means the cross-sectional area of the first wire is twice that of the second wire.

### Step 5: Use the formula for resistance
The resistance \( R \) of a wire is given by:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material (which is the same for both wires).

For the first wire:
\[ R1 = \frac{\rho L}{A1} \]
For the second wire:
\[ R2 = \frac{\rho (2L)}{A2} \]

### Step 6: Substitute the area relationship into the resistance formulas
Substituting \( A1 = 2A2 \) into the equation for \( R1 \):
\[ R1 = \frac{\rho L}{2A2} \]
Now substituting this into the equation for \( R2 \):
\[ R2 = \frac{\rho (2L)}{A2} \]

### Step 7: Find the ratio of the resistances
Now, we can find the ratio of the resistances:
\[ \frac{R1}{R2} = \frac{\frac{\rho L}{2A2}}{\frac{\rho (2L)}{A2}} \]
Canceling \( \rho \) and \( A2 \):
\[ \frac{R1}{R2} = \frac{L}{2L} \cdot \frac{1}{2} = \frac{1}{4} \]

### Conclusion
Thus, the ratio of the resistances \( R1 : R2 \) is:
\[ R1 : R2 = 1 : 4 \]