Two ideal dc voltage sources are connected by electrical conductors that have a resistance...

Two ideal dc voltage sources are connected by electrical conductors that have a resistance of r Ω/m, as shown in Figure. A load having a resistance of R Ω moves between the two voltage sources. Let x equal the distance between the load and the source υ1, and let L equal the distance between the sources.

a) Show that

 

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b) Show that the voltage υ will be minimum when

 

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c) Find x when L = 16 km, υ1 = 1000 V, υ2 = 1200 V,  R = 3.9 Ω, and r = 5 × 10-5 Ω/m.

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d) What is the minimum value of υ for the circuit of part (c)?

Figure