A long power-delivery cable consists of two different conducting alloys fused end-to-end so that current must pass successively through both segments. Segment A has length ℓA=25m, cross-sectional area AA=1.5×10−6m2, and resistivity ρA=2.8×10−8Ω⋅m. Segment B has length ℓB=35m, cross-sectional area AB=2.2×10−6m2, and resistivity ρB=5.1×10−8Ω⋅m. The cable is connected to a regulated 48 V dc supply, and the far end of the cable is attached to a load that draws a constant current of 14 A.
(a) Derive an expression for the total resistance of the composite cable in terms of the given geometric and material parameters, then compute its numerical value.
(b) Determine the current density in each segment of the cable. Briefly justify any mathematical relationships you employ.
(c) Calculate the potential difference that develops across each individual segment when the load is operating.
(d) Engineers wish to reduce the total power dissipated as thermal energy within the cable segments by 40\% while leaving the load current unchanged. Assuming only the cross-sectional area of Segment B can be modified, determine the new area AB′ required to achieve the desired reduction. Provide clear reasoning and show all steps in your calculation.