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:
Show Hint
- \( 9:1 \)
- \( 1:9 \)
- \( 18:1 \)
- \( 6:1 \)
The Correct Option is A
Approach Solution - 1
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 \( \boxed{9:1} \).
Approach Solution -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 \( 9:1 \).
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