When the length of a wire is stretched n times its length, its resistance becomes n2 times the original resistance. Prove it.

Proof: Resistance of a wire stretched n times its length becomes n² times the original resistance

Given:

• Original length of wire = L
• Original radius of wire = r
• Wire is stretched to length = n × L
• Volume of wire remains constant during stretching

Step 1: Understand resistance formula

Resistance, R = ρ × (Length) / (Cross-sectional area) = ρ × L / A

Step 2: Effect of stretching on length and area

Since volume remains constant,
Original volume = New volume
π r² × L = π r’² × (nL)
→ r’² = (r² × L) / (nL) = r² / n
→ New radius, r’ = r / √n

Step 3: Calculate new resistance

New resistance, R’ = ρ × (nL) / (π r’²)
Substitute r’² = r² / n,
R’ = ρ × nL / (π (r² / n)) = ρ × nL × n / (π r²) = n² × (ρ L / π r²)
R’ = n² × R

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Conclusion: When the length of a wire is stretched n times, its resistance becomes n² times the original resistance.