An electric transmission cable is constructed by bundling three long, straight, solid conductors so that they run parallel to each other along their full length of 2.0 km.
• Conductor 1: aluminum, radius 6 mm
• Conductor 2: aluminum, radius 6 mm (identical to Conductor 1)
• Conductor 3: copper, radius 4 mm
At the operating temperature, the resistivity of aluminum is 2.8×10−8Ωm and that of copper is 1.7×10−8Ωm. The conductors are electrically connected in parallel at each end so the bundle acts as a single circuit element.
(a) Derive an algebraic expression for the equivalent resistance Req of the three-conductor bundle in terms of the given geometrical and material parameters.
(b) Calculate the numerical value of Req for the 2.0 km length.
(c) An engineer proposes replacing the bundle with a single solid aluminum wire of the same 2.0 km length and radius 10 mm. Determine the resistance of the proposed single wire and state, with quantitative justification, whether the replacement would increase or decrease the cable’s resistance.