Two wires are made of the same metal. Which of the following statements is NOT correct?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → If the two wires have the same mass and one is twice as long as the other, the longer wire has twice the resistance of the shorter wire.
For a wire of length \( l \) and area of cross-section \( A \), the resistance is \( R=\rho l/A \). The resistivity \( \rho \) depends only on the metal. It does not change with the size or shape of the wire. So for two wires of the same metal, only \( l \) and \( A \) decide the resistance.
Option (1) If the two wires have the same length and one has twice the diameter of the other, the thicker wire has one-fourth the resistance of the thinner wire. — Correct statement. The area of cross-section grows as the square of the diameter. So twice the diameter gives four times the area. For the same length, the resistance becomes one-fourth.
Option (2) If the two wires have the same mass and one is twice as long as the other, the longer wire has twice the resistance of the shorter wire. — Not correct. Wires of the same metal and the same mass have the same volume \( V=lA \). Putting \( A=V/l \) in the resistance gives:
\( \begin{aligned} R &= \frac{\rho l}{A} \\ &= \frac{\rho l^{2}}{V} \end{aligned} \)
The longer wire is also thinner, with half the area. So its resistance is four times that of the shorter wire.
Option (3) If one wire is twice as long as the other and also has twice its area of cross-section, the two wires have equal resistances. — Correct statement. The resistance depends on the ratio \( l/A \), and \( 2l/2A \) is the same as \( l/A \).
Option (4) When a wire is cut into two equal halves, each half has half the resistance of the whole wire but the same resistivity. — Correct statement. Each half has length \( l/2 \) and the same area, so its resistance is \( R/2 \). Its resistivity is still that of the metal.
NCERT (2026-27):
Ohm’s Law – “… In general, then the resistance R is inversely proportional to the cross-sectional area,”
\( \displaystyle R\propto\frac{1}{A} \qquad (3.7) \)
“… and hence for a given conductor”
\( \displaystyle R=\rho\,\frac{l}{A} \)\( \qquad (3.9) \)
“where the constant of proportionality ρ depends on the material of the conductor but not on its dimensions. ρ is called resistivity.”