Answer:
(a) 0.795 m
(b) R = 0.0197 Ω
R = 0.0212 Ω
Explanation:
The resistance of a wire (R) is equal to the resistivity (ρ) times the length (L) divided by the cross sectional area (A). The resistivity of a metal increases with temperature and is a function of the metal's temperature coefficient of resistance (α).
(a) Both wires have the same resistance.
R₁ = R₂
ρ₁ L₁ / A₁ = ρ₂ L₂ / A₂
The diameters are the same, so the cross sectional area is the same.
ρ₁ L₁ = ρ₂ L₂
Resistivity of silver is 1.59×10⁻⁸ Ωm, and resistivity of iron is 1.0×10⁻⁷ Ωm.
(1.59×10⁻⁸ Ωm) (5 m) = (1.0×10⁻⁷ Ωm) L₂
L₂ = 0.795 m
(b) The resistivity of a metal at a temperature T above 20°C is:
ρ = ρ₀ [1 + α (T − 20°C)]
where ρ₀ is the resistivity at 20°C and α is the temperature coefficient of resistance.
For silver:
ρ = ρ₀ [1 + α (T − 20)]
ρ = (1.59×10⁻⁸ Ωm) [1 + (3.8×10⁻³ K⁻¹) (100°C − 20°C)]
ρ = 2.07×10⁻⁸ Ωm
R = ρL/A
R = (2.07×10⁻⁸ Ωm) (5 m) / [¼ π (2.588×10⁻³ m)²]
R = 0.0197 Ω
For iron:
ρ = ρ₀ [1 + α (T − 20)]
ρ = (1.0×10⁻⁷ Ωm) [1 + (5.0×10⁻³ K⁻¹) (100°C − 20°C)]
ρ = 1.4×10⁻⁷ Ωm
R = ρL/A
R = (1.4×10⁻⁷ Ωm) (0.795 m) / [¼ π (2.588×10⁻³ m)²]
R = 0.0212 Ω
