Two wires P and Q are made of the same metal. When the resistance of each wire is plotted against its length, both graphs are straight lines passing through the origin, and the slope of the graph for P is four times that for Q. The ratio of the diameter of P to that of Q is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 1/2 \)
The resistance of a wire is \( R=\rho l/A \). For a wire of fixed material and cross-section, \( R \) is directly proportional to \( l \). So the R–l graph is a straight line through the origin with slope \( \rho/A \). Both wires have the same resistivity \( \rho \), so the steeper graph belongs to the thinner wire.
The ratio of the slopes gives the ratio of the areas:
\( \begin{aligned} \frac{\rho/A_P}{\rho/A_Q} &= 4 \\ \frac{A_Q}{A_P} &= 4 \end{aligned} \)
With \( A=\pi d^{2}/4 \), the area depends on the square of the diameter:
\( \begin{aligned} \frac{d_P^{2}}{d_Q^{2}} &= \frac{A_P}{A_Q} \\ &= \frac{1}{4} \end{aligned} \)
\( \displaystyle \frac{d_P}{d_Q}=\frac{1}{2} \)
So the wire P, whose graph is four times as steep, is half as thick as Q.