Question

A 170-km-long high-voltage transmission line 2.00 cm in diameter carries a steady current of 1,010 A. If the conductor is copper with a free charge density of 8.50 1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable

Answer 1

The number of years is
`t_y = 22.8 \ years`

Step-by-step explanation:

From the question we are told that

The length of the transmission line is
`L = 170 \ km = 170000 \ m`

The diameter of the transmission line is
`d = 2.0 \ cm = 0.02 \ m`

The current which the transmission line carry is
`I = 1,010 \ A`

The charge density of the transmission line is
`j = 8.50 *10^(28 ) \ electron/m^3`

Now the cross-sectional area of the transmission line is mathematically represented as

`A = \pi r^2`

Here r is the radius which is mathematically evaluated as

`r = (d)/(2)`

substituting values

`r = (0.02)/(2)`

`r = 0.01 \ m`

Hence

`A = 3.142 * (0.01)^2`

=>
`A = 3.142 *10^(-4) \ m^2`

Now the drift velocity of electron is mathematically evaluated as

`v = (I)/(j* e * A )`

Where e is the charge on one electron and the values is
`e = 1.60 *10^(-19) \ C`

So

`v = ( 1010)/(8.50 *10^(28)* (1.60 *10^(-19)) * 3.142*10^(-4) )`

`v = 2.363 *10^(-4) \ m/s`

Now the time taken is mathematically evaluated as

`t = (L)/(v )`

substituting values

`t = (170000)/(2.363 *10^(-4) )`

`t = 7.194*10^(8)\ s`

Converting to years

`t_y = (t)/(365\ days * 24 \ hours * 3600\ seconds)`

substituting values

`t_y =(7.194 *10^(8))/(365 *24 * 3600)`

`t_y = 22.8 \ years`