We are given three copper wires of different lengths and different areas of cross-section. Which one of the following would have highest resistivity?
- (a)Copper wire of 50 cm length and 1 mm diameter
- (b)Copper wire of 25 cm length and 0·5 mm diameter
- (c)Copper wire of 10 cm length and 2·0 mm diameter
- (d)All the wires would have same resistivity
Correct — D, all the wires would have same resistivity. Resistivity is a property of the material, not of the piece of material. It is fixed once the substance and the temperature are fixed, and it does not care how long the wire is or how thick. All three wires in this question are copper, so all three share one value of resistivity. What changes with the dimensions is the resistance, through R = ρL/A: a longer wire has more resistance, a thicker one has less. Doing the arithmetic makes the difference vivid. Taking areas as πd²/4, the quantity L/A works out to about 6.4 × 10⁵ per metre for the first wire, 1.27 × 10⁶ for the second and 3.2 × 10⁴ for the third — resistances in the ratio 20 : 40 : 1. Had the question asked which wire has the highest resistance, the answer would have been the second one; it asks for resistivity, and there the three are equal.
- (a)Copper wire of 50 cm length and 1 mm diameter — The longest wire, so it has more resistance than the third — but length raises resistance, not resistivity, and its resistivity is the same copper value as the others.
- (b)Copper wire of 25 cm length and 0·5 mm diameter — The thinnest wire, and the one that would win if the question asked for resistance — halving the diameter quarters the area and quadruples the resistance. Its resistivity is still that of copper.
- (c)Copper wire of 10 cm length and 2·0 mm diameter — Short and thick, so its resistance is the lowest of the three by a factor of twenty or more. Once again the dimensions have moved the resistance and left the resistivity untouched.
Resistance measures how hard a particular object is to push current through; resistivity measures the same difficulty for the substance itself, independent of shape and size. They are tied together by R = ρL/A, where L is the length and A the area of cross-section. Resistivity is measured in ohm-metre, and its only real dependence is on the material and on temperature — in a metal it rises as the wire gets hotter.
The examiner has built the item so that a candidate who reads 'resistance' where the paper says 'resistivity' has three plausible answers to choose between and will pick one of them. The defence is to name the variable before looking at the options: resistivity belongs to copper, so three copper wires cannot differ in it whatever their shape. Two ideas are worth keeping side by side. Resistance is an extensive property, changing when the specimen changes; resistivity is intensive, like density or melting point. And the reason a thin fuse wire melts before the household wiring does is exactly this — same material, different area, different resistance and different heating.
- Resistivity depends only on the material and its temperature, not on the length or thickness of the specimen.
- R = ρL/A — resistance rises with length and falls with area of cross-section.
- The SI unit of resistivity is the ohm-metre.
- The resistivity of a metal increases as its temperature rises; that of a semiconductor falls.
- Copper and aluminium have low resistivity, which is why they are used for wires; nichrome has high resistivity, which is why it is used for heating elements.
The resistances stand in the ratio 20 : 40 : 1; the resistivities stand at 1 : 1 : 1.
- Reading 'resistivity' as 'resistance' — the three wires differ in one and not in the other.
- Assuming a thicker wire must have higher resistivity because it holds more material.
- Forgetting that the diameter is given, not the area — halving the diameter quarters the area.
Either as this comparison of specimens of one material, or as a direct question on which quantity does not change when the wire is stretched or cut.
Which of the following statements are correct about the electrical resistance and resistivity of a wire ? 1. Both quantities depend on the area of cross-section of the wire 2. Both depend on the temperature 3. Resistance of the wire is directly proportional to the resistivity of the wire 4. Resistivity of the wire is directly proportional to the length of the wire Select the correct answer using the code given below :
- (a) 1 and 2
- (b) 1 and 3
- (c) 2 and 3
- (d) 2 and 4
Answer(c) 2 and 3
The rule behind this question spelled out statement by statement. Its true pair says that both quantities vary with temperature and that resistance is proportional to resistivity; its false pair says that resistivity depends on the cross-section and on the length, which is the very mistake the options here invite.
What is the correct sequence of resistivity of silver, nichrome and glass at room temperature?
- (a) Silver < Nichrome < Glass
- (b) Glass < Nichrome < Silver
- (c) Silver < Glass < Nichrome
- (d) Nichrome < Silver < Glass
Answer(a) Silver < Nichrome < Glass
The same quantity asked as a materials comparison. Resistivity separates a good conductor from an insulator across many powers of ten, which is exactly why it cannot separate three wires that are all copper.
- practice — not a real PYQ
A copper wire is cut into two equal halves. The resistivity of each half is
- (a)half the original
- (b)double the original
- (c)the same as the original
- (d)one-fourth the original
Answer(c) the same as the original — cutting changes the length and hence the resistance of each piece, but resistivity belongs to the copper and is untouched.
- practice — not a real PYQ
Two wires of the same material and the same length have diameters in the ratio 1 : 2. The ratio of their resistances is
- (a)1 : 2
- (b)2 : 1
- (c)1 : 4
- (d)4 : 1
Answer(d) 4 : 1 — area goes as the square of the diameter, so the thinner wire has one-quarter the area and four times the resistance.