A conductor having cubold shape has dimensions in ratio 1: 2:5, the ratio of maximum to minimum resistance across two opposite faces is

Options

A. A. 0
B. B. `5:1`
C. C. `25:1`
D. D. `10:1`

Answer

To solve the problem of finding the ratio of maximum to minimum resistance across two opposite faces of a cuboidal conductor with dimensions in the ratio 1:2:5, we can follow these steps:

### Step 1: Define the Dimensions
Let the dimensions of the cuboid be:
- Length (L) = 1x
- Breadth (B) = 2x
- Height (H) = 5x

### Step 2: Identify the Faces
The cuboid has three pairs of opposite faces:
1. Faces with dimensions (Length x Height) = (1x x 5x)
2. Faces with dimensions (Breadth x Height) = (2x x 5x)
3. Faces with dimensions (Length x Breadth) = (1x x 2x)

### Step 3: Calculate Minimum Resistance
The minimum resistance will be across the faces with the smallest area. The area of the faces with dimensions (1x x 2x) is:
\[ A_{min} = 1x \times 2x = 2x^2 \]

Using the formula for resistance:
\[ R = \frac{\rho L}{A} \]
Where:
- \( \rho \) = resistivity of the material
- \( L \) = thickness of the conductor (which we take as the other dimension, here it will be 5x)

Thus, the minimum resistance \( R_{min} \) is:
\[ R_{min} = \frac{\rho (5x)}{2x^2} = \frac{5\rho}{2x} \]

### Step 4: Calculate Maximum Resistance
The maximum resistance will be across the faces with the largest area. The area of the faces with dimensions (2x x 5x) is:
\[ A_{max} = 2x \times 5x = 10x^2 \]

Thus, the maximum resistance \( R_{max} \) is:
\[ R_{max} = \frac{\rho (1x)}{10x^2} = \frac{\rho}{10x} \]

### Step 5: Calculate the Ratio of Maximum to Minimum Resistance
Now, we can find the ratio of maximum resistance to minimum resistance:
\[ \frac{R_{max}}{R_{min}} = \frac{\frac{\rho}{10x}}{\frac{5\rho}{2x}} = \frac{\rho}{10x} \times \frac{2x}{5\rho} \]

The \( \rho \) and \( x \) cancel out:
\[ \frac{R_{max}}{R_{min}} = \frac{2}{50} = \frac{1}{25} \]

Thus, the ratio of maximum to minimum resistance is:
\[ R_{max} : R_{min} = 25 : 1 \]

### Final Answer
The ratio of maximum to minimum resistance across two opposite faces is **25:1**.

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