Questions: Ohm's Law: Microscopic and Macroscopic Forms

Question 1

Two wires are made of the same copper (same resistivity ρ). Wire A is twice as long and half the cross-sectional area of Wire B. How does the resistance of A compare to B?

• A. Wire A has the same resistance as B — same material means same resistance
• B. Wire A has twice the resistance of B
• C. Wire A has 4 times the resistance of B
• D. Wire A has half the resistance of B

Answer: (C)

Resistance R = ρL/A. For Wire A: L_A = 2L, A_A = A/2, so R_A = ρ(2L)/(A/2) = 4ρL/A = 4R_B. Wire A has 4 times the resistance. The common misconception is that same material means same resistance (option A), but R depends on both material (ρ) and geometry (L/A). Doubling length doubles resistance; halving area doubles it again; the effects multiply.

Question 2

In the Drude model, why does current density J depend linearly on the electric field E rather than quadratically or in some other way?

• A. Because charge is defined as current times time, making the relationship linear by definition
• B. Because the mean collision time τ is independent of field strength for ordinary electric fields, so drift velocity — and therefore J — scales linearly with E
• C. Because the number of conduction electrons n doubles when E doubles
• D. Because resistivity ρ is defined as E/J, making the linear relationship tautological

Answer: (B)

The linearity of Ohm's law is not obvious — it follows from the assumption that τ (mean time between collisions) doesn't change with field strength. Under this assumption, drift velocity v_d = qEτ/m scales linearly with E. Since J = nqv_d, J scales linearly with E too, giving J = σE. This is an empirical approximation valid for ordinary field strengths; at very high fields τ can become field-dependent and Ohm's law breaks down.

Question 3

The resistance of a metallic conductor increases when temperature rises because higher temperature increases the number of conduction electrons.

• T. True
• F. False

Answer: False

In metals, the number of conduction electrons n does not significantly change with temperature — metals already have a high, fixed density of free electrons. Instead, higher temperature increases lattice vibrations, which shorten the mean collision time τ. Since σ = nq²τ/m, smaller τ means lower σ and higher resistance. Resistance increases because of more collisions, not more electrons. Semiconductors are opposite: temperature promotes more electrons to the conduction band, increasing n and decreasing resistance.

Question 4

The microscopic Ohm's law J⃗ = σE⃗ applies universally to most materials, while V = IR is mainly an approximation valid for certain geometries.

• T. True
• F. False

Answer: False

Neither form of Ohm's law is universal — both describe a linear regime that holds only for *ohmic* (linear) materials. Diodes, transistors, and many other components are non-ohmic: their J–E relationships are not proportional. The distinction between the microscopic and macroscopic forms is one of generality (J = σE is local; V = IR integrates across a conductor with specific geometry), not of exactness. Both forms break down for non-ohmic materials.

Question 5

Explain why a semiconductor's resistance decreases as temperature increases, while a metal's resistance increases — despite both obeying J = σE with σ = nq²τ/m.

Model answer: The conductivity formula contains two temperature-sensitive quantities: carrier density n and scattering time τ. In metals, n is approximately constant (the conduction band is full), so temperature only affects τ — more lattice vibrations cause more frequent collisions, reducing τ, decreasing σ, and increasing resistance. In semiconductors, the conduction band is mostly empty at low temperatures; increasing temperature promotes electrons across the band gap, dramatically increasing n. This increase in n dominates over the decrease in τ, so σ increases and resistance falls.

Both behaviors follow from the same formula, with different dominant mechanisms. In metals τ is the controlling variable; in semiconductors n is. Temperature coefficient of resistance is positive for metals and negative for semiconductors — a direct consequence. This has practical implications: superconducting metals and cooling improve metallic conductivity, while semiconductor devices conduct better when heated.