Question

7 which of the following aluminum wires would have the greatest resistance
a a wire measuring 10 cm in length and 3 cm in diameter.
b a wire measuring 10 cm in length and 5 cm in diameter.
c a wire measuring 5 cm in length and 3 cm in diameter.
d a wire measuring 5 cm in length and 5 cm in diameter.

Explanation:

Step1: Recall the resistance formula

The resistance formula is \(R =
ho\frac{l}{A}\), where \(
ho\) is the resistivity (constant for aluminum), \(l\) is the length, and \(A=\pi(\frac{d}{2})^2\) is the cross - sectional area (\(d\) is the diameter).

Step2: Analyze the effect of length and diameter on resistance

• Length (\(l\)): Resistance is directly proportional to length (\(R\propto l\)). Longer length means higher resistance.
• Diameter (\(d\)): Resistance is inversely proportional to the cross - sectional area \(A\) (\(R\propto\frac{1}{A}\propto\frac{1}{d^{2}}\)). Smaller diameter means higher resistance.

Step3: Compare the options

• Option A: \(l = 10\space cm\), \(d = 3\space cm\)
• Option B: \(l = 10\space cm\), \(d = 5\space cm\). Larger \(d\) than A, so \(R_B
• Option C: \(l = 5\space cm\), \(d = 3\space cm\). Smaller \(l\) than A, so \(R_C
• Option D: \(l = 5\space cm\), \(d = 5\space cm\). Smaller \(l\) and larger \(d\) than A, so \(R_D

Answer:

A. A wire measuring 10 cm in length and 3 cm in diameter.