Question

A block has dimensions 1 cm, 2 cm, and 3 cm. The ratio of the maximum resistance to minimum resistance between any pair of opposite faces of the block is:

• A. \( 9:1 \)
• B. \( 1:9 \)
• C. \( 18:1 \)
• D. \( 6:1 \)

Hint:

To find the resistance ratio of a rectangular block, always use \( R = \rho \frac{L}{A} \) and compare the maximum and minimum resistances by choosing the longest and shortest current flow paths.

Answer 1

The Correct Option is A

A block has dimensions \(1 \, \text{cm} \), \(2 \, \text{cm} \), and \(3 \, \text{cm}\). The resistance \( R \) between opposite faces is given by:

\[ R = \rho \frac{L}{A} \]

where:

• \( \rho \) is the resistivity of the material
• \( L \) is the length of the current path
• \( A \) is the cross-sectional area perpendicular to the current

Step-by-step Calculation:

• First pair of faces: Area = \(2 \times 3 = 6 \, \text{cm}^2\), Length = \(1 \, \text{cm}\) 
  \[ R_1 = \rho \frac{1}{6} \]
• Second pair of faces: Area = \(1 \times 3 = 3 \, \text{cm}^2\), Length = \(2 \, \text{cm}\) 
  \[ R_2 = \rho \frac{2}{3} \]
• Third pair of faces: Area = \(1 \times 2 = 2 \, \text{cm}^2\), Length = \(3 \, \text{cm}\) 
  \[ R_3 = \rho \frac{3}{2} \]

Final Step:

Maximum resistance: \( R_{\text{max}} = R_3 = \rho \frac{3}{2} \) 
Minimum resistance: \( R_{\text{min}} = R_1 = \rho \frac{1}{6} \)

\[ \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{\rho \cdot \frac{3}{2}}{\rho \cdot \frac{1}{6}} = \frac{3}{2} \cdot \frac{6}{1} = 9 \]

Conclusion:

The ratio of the maximum resistance to the minimum resistance is \( \boxed{9:1} \).

Answer 2

Step 1: Understanding Resistance of a Rectangular Block The resistance \( R \) of a conducting block is given by the formula: \[ R = \rho \frac{L}{A} \] where: - \( \rho \) is the resistivity of the material, - \( L \) is the length along which current flows, - \( A \) is the cross-sectional area perpendicular to the current. 
Step 2: Finding Maximum and Minimum Resistances Given dimensions: \( 1 \) cm, \( 2 \) cm, and \( 3 \) cm. - Case 1 (Maximum Resistance): - Current flows along the longest dimension (3 cm). - Cross-sectional area = \( 1 \times 2 = 2 \) cm\(^2\). - Resistance: \[ R_{\max} = \rho \frac{3}{2} \] - Case 2 (Minimum Resistance): - Current flows along the shortest dimension (1 cm). - Cross-sectional area = \( 2 \times 3 = 6 \) cm\(^2\). - Resistance: \[ R_{\min} = \rho \frac{1}{6} \] 
Step 3: Finding the Ratio \[ \frac{R_{\max}}{R_{\min}} = \frac{\rho \frac{3}{2}}{\rho \frac{1}{6}} \] \[ = \frac{3}{2} \times \frac{6}{1} \] \[ = \frac{18}{2} = 9:1 \] Thus, the correct ratio of maximum to minimum resistance is \( 9:1 \).