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

Options

A. 1:25
B. `5:1`
C. `25:1`
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 of the cuboid
Let the dimensions of the cuboid be:
- Length (L) = 5x
- Width (W) = 2x
- Height (H) = 1x

### Step 2: Understand the formula for resistance
The resistance (R) of a conductor is given by the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) = resistivity of the material (constant for this problem)
- \( L \) = length of the conductor in the direction of current flow
- \( A \) = cross-sectional area perpendicular to the current flow

### Step 3: Calculate maximum resistance
To find the maximum resistance, we need to maximize the length and minimize the cross-sectional area:
- Maximum Length (L_max) = 5x (along the length)
- Minimum Area (A_min) = Width × Height = 2x × 1x = 2x²

Using the formula for resistance:
\[ R_{max} = \frac{\rho (5x)}{2x^2} = \frac{5\rho}{2x} \]

### Step 4: Calculate minimum resistance
To find the minimum resistance, we need to minimize the length and maximize the cross-sectional area:
- Minimum Length (L_min) = 1x (along the height)
- Maximum Area (A_max) = Width × Length = 2x × 5x = 10x²

Using the formula for resistance:
\[ R_{min} = \frac{\rho (1x)}{10x^2} = \frac{\rho}{10x} \]

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

### Step 6: Simplify the ratio
The \( \rho \) and \( x \) terms cancel out:
\[ \frac{R_{max}}{R_{min}} = \frac{5}{2} \times 10 = \frac{50}{2} = 25 \]

### Conclusion
The ratio of maximum to minimum resistance across two opposite faces of the cuboidal conductor is:
\[ \frac{R_{max}}{R_{min}} = 25:1 \]