How Long Does an Electron Take to Travel the Length of a High-Voltage Cable?

[AznBoi]

Homework Statement

A 200-km-long high-voltage transmission line 2 cm in diameter carries a steady current of 1000 A. If the conductor is copper with a free charge density of 8.5 x 10^28 electrons per cubic meter, how many years does it take one electron to travel th full length of the cable?

The Attempt at a Solution

I know that you need to use this equation: $$R=p\frac {l}{A}$$ and I think I will also need this equation: $$I= \frac {\Delta {Q}}{\Delta t}$$, but I'm not entirely sure.

The thing I most confused about is the "density of 8.5 x 10^28 electrons per cubic meter. Is that a volume? Are you suppose to use: $$I= \frac {\Delta {Q}}{\Delta t}$$ to find the time (years)?? I'm stuck from here on out. It'd be great if you could give me some hints as to how to continue. Thanks in advance!

 

[berkeman]

I don't think you need the resistivity equation, just use the charge density equation and the equation for current expressed in terms of charge flux.

 

[m2003]

I would approach this problem by first understanding the concept of current and drift speed. Current is the rate at which charge flows through a conductor, while drift speed is the average speed at which charges move through the conductor. In this case, the steady current of 1000 A tells us the rate at which charge is flowing through the transmission line, but it does not give us information about the individual electron's speed.

To calculate the time it takes for one electron to travel the full length of the cable, we will need to use the equation for current, which is I = ΔQ/Δt, where I is the current, ΔQ is the change in charge, and Δt is the time interval. We can rearrange this equation to solve for the time interval, which will give us the time it takes for one electron to travel the full length of the cable.

Next, we need to determine the change in charge, ΔQ, which is equal to the product of the charge density (ρ) and the volume of the conductor (V). The charge density, or the number of electrons per unit volume, is given to us as 8.5 x 10^28 electrons per cubic meter. We can calculate the volume of the conductor using its length and diameter, and then plug in the values for ρ and V into the equation to solve for ΔQ.

Once we have the value for ΔQ, we can plug it into the equation for current and solve for Δt, which will give us the time it takes for one electron to travel the full length of the cable. From there, we can convert the time interval into years to answer the question.

In summary, to solve this problem as a scientist, we need to understand the concept of current and drift speed, and use the given information to calculate the change in charge and time interval. It is important to carefully consider the units and use the appropriate equations to solve the problem accurately.