CDS (II) 2022 General Knowledge · Question 28 of 120

Correct — B, R/4. The resistance of a uniform wire is R = rho x l divided by A, where rho is the resistivity of the material, l the length and A the area of cross-section. Same metal means rho is unchanged; same length means l is unchanged; so the whole answer sits in A. The cross-section of a wire is a circle, and the area of a circle goes as the square of its diameter, A = pi x d squared divided by 4. Double the diameter and the area becomes four times as large, not twice as large. Resistance is inversely proportional to area, so it falls to one quarter of what it was: R divided by 4. The physical picture is the same one — a thicker wire offers four times as many parallel paths for the drifting electrons, and four identical resistors in parallel give a quarter of the resistance of one.

• (a)R — The answer if resistance depended only on the material and the length. It does not; geometry enters through the area of cross-section, and the area has changed by a factor of four.
• (c)R/2 — The single commonest slip on this item. It comes from treating resistance as inversely proportional to the diameter instead of to the area — forgetting to square. Doubling the diameter quadruples the area.
• (d)2R — Gets the direction wrong as well as the size. Making a wire thicker cannot make it harder for current to pass; a doubling of resistance would follow from doubling the length, not the diameter.

Resistance measures how strongly a conductor opposes the flow of charge through it, and for a uniform conductor it separates cleanly into a material part and a shape part. The material part is the resistivity rho, a property of the substance and its temperature — about 1.6 x 10 to the power minus eight ohm-metre for silver, 1.7 for copper, 2.8 for aluminium, and enormously larger for insulators. The shape part is length divided by area of cross-section. Doubling the length doubles the resistance; doubling the area halves it. Resistivity itself does not change when you cut, stretch or thicken a wire — that is what makes it useful.

The trap in this item is purely geometric, and it is set deliberately: the stem gives the diameter, not the area. A student who has memorised 'resistance is inversely proportional to the area of cross-section' but reads the word diameter as though it meant area lands on R/2 and loses the mark. Get into the habit of converting the datum you are given into the quantity the formula wants before doing anything else — diameter to area, and remember the square. The same square catches people in reverse when a wire is stretched: stretching to double the length also halves the area, because the volume of metal is fixed, so the resistance goes up by a factor of four rather than two. One is the mirror of the other, and CDS and NDA have used both versions.

• R = rho x l divided by A, where rho is resistivity, l is length and A is the area of cross-section.
• For a circular wire A = pi x d squared divided by 4, so resistance is inversely proportional to the square of the diameter.
• Doubling the diameter quarters the resistance; halving the diameter multiplies it by four.
• Resistivity is a property of the material and its temperature, and is unchanged by the wire's dimensions; its unit is the ohm-metre.
• If a wire is stretched to n times its length at constant volume, its resistance becomes n squared times the original.

• Reading 'diameter' as if it were 'area' and forgetting to square — the slip that produces the tempting R/2.
• Assuming resistivity changes when the wire's shape changes; it does not.
• Confusing the thickening case with the stretching case, where length and area both change together.

As a proportionality numerical — change one dimension of a wire and report the new resistance — or as its stretched-wire twin.

• practice — not a real PYQ

  A uniform wire of resistance R is stretched so that its length becomes twice the original, the volume of the metal remaining the same. Its new resistance is

  • (a)2R
  • (b)R/2
  • (c)4R
  • (d)R/4

  Answer(c) 4R — the length doubles and the area of cross-section halves, and resistance goes as length divided by area, so it rises by a factor of 2 x 2.

• practice — not a real PYQ

  Which one of the following quantities of a metal wire remains unchanged when the wire is drawn into a thinner one of the same material?

  • (a)Resistance
  • (b)Resistivity
  • (c)Area of cross-section
  • (d)Length

  Answer(b) Resistivity — it is a property of the material and its temperature, not of the wire's dimensions.