The specific resistance of the material of a wire is `'rho'` and its volume is `3m^3` and its resistance is `3 Omega `. The length of the wire will be

Options

A. `(1)/(( sqrt(rho))`
B. `(3)/( sqrt((rho ))`
C. `( sqrt(3))/(rho)`
D. `(rho)/( sqrt(3))`

Answer

To find the length of the wire given its specific resistance (ρ), volume (V), and resistance (R), we can follow these steps:

### Step 1: Understand the relationship between volume, area, and length
The volume (V) of the wire can be expressed in terms of its cross-sectional area (A) and length (L):
\[ V = A \times L \]
Given that the volume \( V = 3 \, m^3 \), we can express the area in terms of length:
\[ A = \frac{V}{L} = \frac{3}{L} \]

### Step 2: Use the formula for resistance
The resistance (R) of the wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
Substituting the expression for area from Step 1 into this formula gives:
\[ R = \frac{\rho L}{\frac{3}{L}} \]
This simplifies to:
\[ R = \frac{\rho L^2}{3} \]

### Step 3: Substitute the known resistance value
We know that the resistance \( R = 3 \, \Omega \). Therefore, we can set up the equation:
\[ 3 = \frac{\rho L^2}{3} \]

### Step 4: Solve for \( L^2 \)
To isolate \( L^2 \), multiply both sides by 3:
\[ 9 = \rho L^2 \]
Now, rearranging gives:
\[ L^2 = \frac{9}{\rho} \]

### Step 5: Take the square root to find \( L \)
Taking the square root of both sides to find the length \( L \):
\[ L = \sqrt{\frac{9}{\rho}} \]
This simplifies to:
\[ L = \frac{3}{\sqrt{\rho}} \]

### Final Result
Thus, the length of the wire is:
\[ L = \frac{3}{\sqrt{\rho}} \]

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