An underground cable 11 km long extends east to west and consists of two parallel wires,...
An underground cable 11 km long extends east to west and
consists of two parallel wires, each of which has resistance 13
Ω/km. An electrical short develops at distance x from the
west end when a conducting path of resistance R connects
the wires (see the figure). The resistance of the wires and the
short is 110 Ω when measured from the east end and 220 Ω when
measured from the west end. What are
(a) x and
(b) R?
Homework Answers
west - - - - - - east
A____P______X
B____Q______Y
The 2 parallel wires are AX and BY.
The short is a resistance R between P and Q.
Working in Ω and km:
AP = BQ = x, the unknown distance.
PX = QY = 11-x (since total distance AX=BY=11km)
The path XPQY has resistance 110 ohms. It consists of PX (resistance 13(11-x)), PQ (resistance R) and QY (resistance 13(11-x)) in series.
13*(11-x) + R + 13(11-x) = 110
Rearranged gives
26x – R = 176 (equation 1)
The path APQB has resistance 220 ohms. It consists of AP (resistance 13x, PQ (resistance R) and BQ (resistance 13x) in series.
13x + R + 13x = 220
26x + R = 220 (equation 2)
(a) To solve them simultaneously, Adding equations 1 and 2:
52x = 396
x = 7.62 km
(b) Thus, from equation 2:
R = 220 -26x
. . = 220 – 26*(7.62)
. . = 22 Ω
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