Questions: Resistivity and Conductivity of Materials

Question 1

Two resistors are both measured at R = 10 Ω. One is made of copper (ρ ≈ 1.7×10⁻⁸ Ω·m) and the other of silicon (ρ ≈ 640 Ω·m). A student concludes they are made of equivalent conducting materials since their resistances are identical. What is wrong with this reasoning?

• A. The student is right — resistance is the only property that matters for circuit behavior, so equivalent resistance means equivalent material
• B. Resistance depends on both material resistivity and geometry via R = ρL/A; the two resistors have the same R but vastly different resistivities — their dimensions must differ by many orders of magnitude to compensate
• C. The student should compare conductance, not resistance, when evaluating material quality
• D. Resistivity only differs between metals and non-metals, not within the same category

Answer: (B)

R = ρL/A separates material properties (ρ) from geometry (L/A). A tiny copper wire and a massive silicon block can have the same resistance if their L/A ratios compensate for the ~10²⁰ difference in resistivity. Same resistance tells you about the circuit element as a whole; resistivity tells you something intrinsic about the material regardless of shape. This is why engineers select materials based on resistivity, not resistance — a material spec is geometry-independent.

Question 2

A copper wire of length L and cross-sectional area A has resistance R. It is replaced by a copper wire of length 2L and cross-sectional area A/2. What is the new resistance?

• A. R — resistivity is a material property, so resistance is unchanged when using the same material
• B. 2R — only the length doubled, which doubles resistance
• C. 4R — resistance scales as L/A; doubling L multiplies R by 2, and halving A multiplies R by another factor of 2, giving 4R total
• D. R/2 — the effects of longer length and smaller area cancel each other out

Answer: (C)

From R = ρL/A, resistance is proportional to L and inversely proportional to A. Doubling L doubles R (longer path, more resistance). Halving A also doubles R (narrower cross-section, less room for current). Both changes increase resistance, so they compound: new R = ρ(2L)/(A/2) = 4ρL/A = 4R. A common error is thinking the changes 'cancel' because one increases length and one decreases area — but both changes increase resistance, they don't oppose each other.

Question 3

A piece of copper wire and a silicon semiconductor can have the same measured resistance even though their resistivities differ by a factor of roughly 10²⁰.

• T. True
• F. False

Answer: True

Yes — because R = ρL/A, the geometry (the ratio L/A) can compensate for any difference in ρ. To match the resistance of a 1 cm copper wire, a silicon sample would need an L/A ratio about 10²⁰ times smaller (much shorter or much thicker). While impractical at extremes, the principle is sound: resistance is a property of the specific physical object, while resistivity is a property of the material. Two objects with the same R can be made of completely different materials.

Question 4

When temperature rises, both metals and semiconductors become better conductors, because higher temperatures increase the kinetic energy of electrons and allow them to move more freely.

• T. True
• F. False

Answer: False

Metals and semiconductors behave oppositely with temperature. In metals, higher temperatures increase atomic vibrations, which scatter conduction electrons more frequently — resistivity *increases* (conductivity decreases). In semiconductors, higher temperatures excite more electrons across the band gap into the conduction band, increasing the number of charge carriers — resistivity *decreases* (conductivity increases). The temperature dependence of resistivity is a key diagnostic for identifying whether a material behaves as a metal or semiconductor.

Question 5

Explain why resistivity is considered a 'material property' while resistance is not. What does the formula R = ρL/A reveal about the relationship between them?

Model answer: Resistivity ρ is intrinsic to the material — it is the same for every sample of copper, regardless of whether you have a thin wire or a thick rod, a short piece or a long one. Resistance R, on the other hand, depends on the geometry of the specific object: longer conductors have higher resistance (R ∝ L), and wider conductors have lower resistance (R ∝ 1/A). The formula R = ρL/A cleanly separates these: ρ carries the material information, while L/A carries the geometric information. This is why engineers choose materials based on resistivity (a material spec), then calculate resistance based on the object's dimensions.

This separation is exactly analogous to distinguishing material density from an object's mass — density is intrinsic, mass depends on how much material you have. Resistivity is the 'density of electrical resistance' in a sense: it tells you the resistance per unit length per unit cross-section.