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
Correct — C, 2 and 3. Take the statements one at a time against R = ρl/A, where ρ is the resistivity. Statement 1 fails, because although resistance does depend on the area of cross-section, resistivity does not — it is a characteristic property of the material and is unchanged by how thick or thin you draw the wire. Statement 2 holds, because both the resistance and the resistivity of a material vary with temperature. Statement 3 holds, because with l and A fixed the equation makes R directly proportional to ρ. Statement 4 fails for the same reason as statement 1 — resistance grows with length, resistivity does not. Only 2 and 3 survive.
- (a)1 and 2 — Statement 2 is correct, but statement 1 is not. Only resistance depends on the area of cross-section; resistivity is fixed by the material and is unaffected by the wire's thickness.
- (b)1 and 3 — Statement 3 is correct, but statement 1 fails for the same reason as above — cutting a wire thinner raises its resistance while leaving its resistivity exactly where it was.
- (d)2 and 4 — Statement 2 is correct, but statement 4 is not. Resistance is directly proportional to length; resistivity is not, and a longer piece of the same copper wire has exactly the same resistivity as a shorter one.
Resistance is a property of a particular piece of wire, so it changes if you change the geometry of that piece. Resistivity is a property of the substance the wire is made of, so it survives any reshaping of the piece. The equation R = ρl/A links them, and temperature is the one influence that acts on both — it changes the material's resistivity, and through it the resistance of every piece made of that material.
The examiner has built the item around a single confusion, that anything true of resistance must also be true of resistivity. Two of the four statements are simply that confusion written out. The test to apply to each statement is whether the quantity would change if you took the same wire and redrew it — if it would, it is resistance; if it would not, it is resistivity. Temperature is the exception that belongs to both, and remembering that exception is what makes statement 2 a keeper. The proportionality in statement 3 is safe to accept because it is the equation read directly, with the geometry held fixed.
- The resistance of a uniform metallic conductor is directly proportional to its length and inversely proportional to its area of cross-section, giving R = ρl/A.
- Resistivity ρ is a characteristic property of the material and its SI unit is the ohm metre.
- Both the resistance and the resistivity of a material vary with temperature.
- At 20 degrees C copper's resistivity is about 1.62 × 10⁻⁸ ohm m while nichrome's is about 100 × 10⁻⁶ ohm m, which is why nichrome and not copper is used in heating elements.
Only statements 2 and 3 survive, which is option (c).
- Carrying a property of resistance across to resistivity without checking it.
- Assuming resistivity cannot change at all because it is called a material property; it does change with temperature.
As a multi-statement item like this one, as a numerical on a stretched or cut wire, or as a straight question on the unit of resistivity.
Which one of the following correctly represents the SI unit of resistivity?
- (a) Ω
- (b) Ω/m
- (c) Ω cm
- (d) Ω m
Answer(d) Ω m
The same equation approached through its units. Rearranging R = ρl/A gives ρ = RA/l, whose unit is ohm metre — and a candidate who can do that rearrangement can also see at once why resistivity cannot depend on l or A.
Which one of the following statements regarding Ohm's law is not correct?
- (a) Ohm's law is an assumption that current through a conductor is always directly proportional to the potential difference applied to it.
- (b) A conducting device obeys Ohm's law when the resistance of a device is independent of magnitude and polarity of applied potential difference.
- (c) A conducting material obeys Ohm's law when the resistance of material is independent of the magnitude and direction of applied electric field.
- (d) All homogeneous materials obey Ohm's law irrespective of whether the field is within range or strong.
Answer(d) All homogeneous materials obey Ohm's law irrespective of whether the field is within range or strong.
The law that sits behind the resistance in this item, tested for its limits. NDA likes multi-statement physics of exactly this shape, where three careful statements are true and one overreaches.
- practice — not a real PYQ
A wire of resistance R is stretched uniformly until its length is doubled, the material remaining the same. Its new resistance is
- (a)R/2
- (b)2R
- (c)4R
- (d)R
Answer(c) 4R — doubling the length halves the area, and R = ρl/A then gives four times the original resistance.
- practice — not a real PYQ
The SI unit of electrical resistivity is
- (a)ohm
- (b)ohm per metre
- (c)ohm metre
- (d)ohm per square metre
Answer(c) ohm metre — it follows from ρ = RA/l.