Final answer:
To calculate the time it takes one electron to travel the length of a high-voltage transmission line, you can find the drift velocity of electrons using the current, charge density, and cross-sectional area of the copper wire, then divide the length of the wire by this velocity and convert the time from seconds to years.
Step-by-step explanation:
To determine how many years it takes one electron to travel the full length of a 160 km-long high-voltage transmission line, we can calculate the drift velocity of electrons in a copper conductor carrying a steady current. First, we use the current (I) and the number of free charge carriers (n) to find the drift velocity (vd) using the relation I = nqAvd, where I is the current, n is the charge density, q is the charge of an electron, A is the cross-sectional area of the wire, and vd is the drift velocity. Then, we calculate the time it takes for an electron to travel the full length of the cable and convert this time into years using the number of seconds in a year.
Given:
- Current, I = 1,080 A
- Diameter of the wire, d = 2.00 cm
- Free charge density, n = 8.50 x 1028 electrons/m3
- Charge of an electron, q = 1.60 x 10-19 C
The cross-sectional area A of a wire with diameter d is given by A = \(\pi r^2\), where r is the radius of the wire. Thus, the area A = \(\pi(0.01 m)^2\).
The drift velocity vd can be calculated using the formula vd = I / (nqA). With all values known, we can compute vd. To find the time (t) it takes for an electron to travel the full length of the cable, we use t = length/vd, where the length of the cable is 160,000 meters. Finally, to convert the time from seconds to years, we divide by the number of seconds in a year, 3.156 x 107 s/year.
Once the calculation is done, we get the time in years, which is the answer to the question as to how long it takes one electron to travel the full length of the cable when a steady current is applied.