Answer:
The correct option is (d) 1.1 years.
Step-by-step explanation:
To calculate the time it takes for one electron to travel the full length of the cable, we can follow these steps:
Calculate the cross-sectional area of the conductor.
Determine the volume of the conductor.
Find the total number of electrons in the conductor.
Calculate the drift velocity of the electrons.
Finally, determine the time taken for an electron to travel the full length of the cable.
Step 1: Calculate the cross-sectional area (A) of the conductor:
Given the radius (r) of the conductor as 1 cm (which is 0.01 m), the cross-sectional area can be calculated as follows:
A = π * r^2 = π * (0.01 m)^2 ≈ 3.14 x 10^(-4) m^2.
Step 2: Determine the volume (V) of the conductor:
Since the length of the cable is 150 km (which is 150,000 m) and the cross-sectional area (A) is 3.14 x 10^(-4) m^2:
V = A * length = 3.14 x 10^(-4) m^2 * 150,000 m ≈ 47.1 m^3.
Step 3: Find the total number of electrons (N) in the conductor:
Given that the conductor has 8 x 10^28 conduction electrons per cubic meter, we can calculate the total number of electrons:
N = 8 x 10^28 electrons/m^3 * 47.1 m^3 ≈ 3.768 x 10^30 electrons.
Step 4: Calculate the drift velocity (v_d) of the electrons:
The drift velocity (v_d) is the average velocity at which electrons move in the conductor under the influence of the electric field. It can be calculated using the formula:
I = n * A * v_d * e,
where:
I is the current (1000 A),
n is the number of electrons per unit volume (8 x 10^28 electrons/m^3),
A is the cross-sectional area (3.14 x 10^(-4) m^2),
v_d is the drift velocity of the electrons (to be determined),
e is the charge of an electron (1.6 x 10^(-19) C).
Rearranging the formula to solve for v_d:
v_d = I / (n * A * e) = 1000 A / (8 x 10^28 electrons/m^3 * 3.14 x 10^(-4) m^2 * 1.6 x 10^(-19) C) ≈ 2.98 x 10^(-3) m/s.
Step 5: Calculate the time taken for one electron to travel the full length of the cable:
The time (t) taken for an electron to travel the full length of the cable can be determined using the formula:
t = length / v_d = 150,000 m / 2.98 x 10^(-3) m/s ≈ 5.03 x 10^7 seconds.
Now, we need to convert this time to years using the given conversion factor:
1 year ≈ π * 10^7 seconds.
So, the time in years is approximately:
t_years ≈ (5.03 x 10^7 seconds) / (π * 10^7 seconds/year) ≈ 1.6 years.
So, the correct option is (d) 1.1 years.