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A wire made of material A has twice the length and half the cross-sectional area

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Question: A wire made of material A has twice the length and half the cross-sectional area of a wire made of material B. If the resistivity of A is ρ, what is the resistance of wire A in terms of the resistance of wire B?

Options:

  1. 2R
  2. 4R
  3. R/2
  4. R/4

Correct Answer: 4R

Solution: 

Resistance R = ρ(L/A). For wire A, R_A = ρ(2L/(A/2)) = 4ρ(L/A) = 4R_B.

A wire made of material A has twice the length and half the cross-sectional area

Practice Questions

Q1
A wire made of material A has twice the length and half the cross-sectional area of a wire made of material B. If the resistivity of A is ρ, what is the resistance of wire A in terms of the resistance of wire B?
  1. 2R
  2. 4R
  3. R/2
  4. R/4

Questions & Step-by-Step Solutions

A wire made of material A has twice the length and half the cross-sectional area of a wire made of material B. If the resistivity of A is ρ, what is the resistance of wire A in terms of the resistance of wire B?
Correct Answer: 4R_B
  • Step 1: Understand the formula for resistance, which is R = ρ(L/A), where R is resistance, ρ is resistivity, L is length, and A is cross-sectional area.
  • Step 2: Identify the properties of wire A: it has twice the length of wire B, so L_A = 2L_B, and it has half the cross-sectional area of wire B, so A_A = A_B / 2.
  • Step 3: Substitute the values for wire A into the resistance formula: R_A = ρ(L_A / A_A).
  • Step 4: Replace L_A and A_A with their expressions in terms of wire B: R_A = ρ((2L_B) / (A_B / 2)).
  • Step 5: Simplify the expression: R_A = ρ(2L_B * (2/A_B)) = ρ(4L_B / A_B).
  • Step 6: Recognize that R_B = ρ(L_B / A_B), so we can express R_A in terms of R_B: R_A = 4 * R_B.
  • Resistance Calculation – Understanding how to calculate resistance using the formula R = ρ(L/A), where ρ is resistivity, L is length, and A is cross-sectional area.
  • Proportional Relationships – Recognizing how changes in length and area affect resistance, particularly how doubling length and halving area interact.
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