00:01
To solve this problem, we identify the things that we're given, and we're given the diameter of the wires, which is 1 millimeter or 10 to minus 3 meters.
00:12
The resistivity, or the resistance, rather, 1 -on.
00:17
The materials we're asked to work with is gold, copper, and aluminum, and the resistivity can be found in table 25 .1.
00:26
And in part a of the problem, we're asked to determine the length of each wire that will give us a resistance of one for a diameter of one millimeter of each of the wires.
00:43
So in order to solve this problem, or to solve part a, we use equation 2510, 25 .10.
00:56
And this equation tells us that the resistance is equal to the resistivity times the length of the wire divided by the cross -sectional area in here we can just rearrange this equation so that we can isolate the length so the length is equal to the resistance times a cross -sectional area divided by the resistivity so we're given the resistance, we're given the resistivity in table 25 .1, and we can easily calculate the cross -sectional area by remembering that the wire is a cylinder and it has a circle for cross -sectional area.
01:50
So that's just pi r squared, which is pi times one -half the diameter squared.
02:01
So now we have all the relevant pieces of information.
02:06
Find the length of each of the wires that we're asked to work with.
02:12
I'll do this, i'll calculate the actual numbers on the next page.
02:20
So the length of, well i'll just rewrite the equation here so we know what we'll work with.
02:26
So the length is the resistance times the cross -sectional area divided by the resistivity.
02:35
So this is our main equation here.
02:39
So so the length of the gold wire that we need is 1 ome times the cross -sectional area, which is 1 half times 1 times 10 to the minus 3 meters squared times pi.
03:06
So this portion is the cross -sectional area divided by the resistivity of gold, which is 2 .4.
03:19
2 .44 times 10 to the minus 8 ome meter.
03:26
And you'll notice that this equation gives you the right dimensions for the length, which is meters, because the ohms cancel out, and a meter from the denominator cancel out one of the meters from the numerator.
03:43
And you're a laugh with just meters.
03:47
So the length of gold, after you run these numbers, you'll find that it is 32 .17 meters.
04:03
In a similar way, we compute the length of the copper and the aluminum wire.
04:11
So the only thing that's really changing here will be the resistivity of the materials because we're told that we want the same resistance and we're using the same diameter for each of the wires...