A 1-meter-long wire consists of an inner copper core with a radius of 1.0 mm and an outer aluminum sheathe, which is 1.0 mm thick, so the total radius of the wire is 2.0 mm. The resistivity of copper is 1.7×10−8 Ω·m and that of aluminum is 2.8×10−8 Ω·m. What is the total resistance of the wire? (Hint, are the two different materials in series or in parallel with each other?)

Explanation:

Given:

Length, L = 1 m

radius, rc = 1.0 mm

Area of inner copper, Ac = pi × (0.001)^2

= 3.142 × 10^-6 m^2

Thickness, t = 1.0 mm

Total radius of the wire, rt = 2.0 mm

Area of outer aluminum sheathe, Aa = area of total wire, At - area of copper core, Ac

Area of total wire = pi × (0.002)^2

= 1.26 × 10^-5 m^2

Aa = 1.26 × 10^-5 - 3.142 × 10^-6

= 9.42 × 10^-6 m^2

Resistivity of copper, Dc = 1.7×10−8 Ω·m

Resistivity of aluminum, Da = 2.8×10−8 Ω·m

D = (R × A)/L

Rc = (Dc × L)/Ac

= (1.7×10−8 × 1)/3.142 × 10^-6

= 5.41 × 10^-3 Ω

Ra = (2.8×10−8 × 1)/9.42 × 10^-6

= 2.97 × 10^-3 Ω

Since both wires are connected at the same time to the voltage supply, therefore,

1/Rt = 1/Ra + 1/Rc

= 1/2.97 × 10^-3 + 1/5.41 × 10^-3

= 521.54

Rt = 1.92 × 10^-3 Ω