In this video, we're going to talk about resistivity and resistance of wires. So what equation can we use to calculate the resistance of a wire? The resistance of a wire is equal to the resistivity, represented by the symbol rho, times the length of the wire, divided by the area. So here's a question for you. Which wire will have more resistance?
A short wire, or a long wire? What would you say? It turns out that the long wire has more resistance.
Now, why is that the case? Well, as you increase the length of a wire, the resistance will increase, if you look at the formula. Notice that L is on the top of the numerator.
Whenever you increase the numerator of a fraction, the entire value of the fraction increases. So as you increase the length of a wire, the resistance of that wire will increase. Now here's another question for you. Which wire has more resistance?
A thin wire or a thick wire? And let's say they have the same length. Which one has more resistance?
It turns out that the thin wire has more resistance than the thick wire. Looking at the equation, the area is on the bottom. As you increase the cross sectional area of a wire, the resistance will decrease. Whenever you increase the value of the denominator, the value of the whole fraction decreases.
Now if you think about it, the reason why the thick wire has less resistance is because there's more space for the electrons to move. And because there's more space, that conductor can carry more electrons. And so more electrons can flow. If there's more current, there's less resistance given the same.
same voltage. Now you can think of it as a highway. The thin wire, think of it as a one-lane highway. There's not many cars that can pass through a one-lane highway, so there's a lot more resistance in a thin wire.
But in the thick wire, imagine it as a seven-lane highway. More cars can flow through a seven-lane highway, and so there's less resistance. In the case of length, the longer the road, the more time it's going to take to get through it.
So, if you increase the length, the resistance will increase. Now, what about P, the resistivity? What does that tell us?
That is a property of the metal itself, or the material that conducts electricity. In the case of a good conductor, like a metal, the resistivity is in the range of 10 to the minus 8. It's pretty low. Metals have low resistance, which means that they're very good conductors. connectors are silver and copper.
Silver has a resistivity of 1.59 times 10 to negative 8 and copper is about 1.6. 6 8 times 10 to negative 8. Based on this, which of these two metals is a better conductor, silver or copper, given their resistivity values? Now copper has higher resistivity than silver so therefore copper has more resistance so if copper has more resistance then it's going to conduct electricity a little bit less efficient than silver So resistivity and conductivity, they're inversely related. Because copper has a higher resistivity, it's less conductive. And because silver has a lower resistivity, it's more conductive.
So silver is a better conductor than copper. So materials with low resistivity means that they're good electrical conductors. Metals typically have a resistivity in this order, 10 to negative 8. Semiconductors or semi metals like carbon graphite, germanium, and silicon, the resistivity is much higher. In the case of carbon graphite, it's about anywhere between 3 to 60 times 10 to minus 5. So it's much higher than a typical metal, but it still conducts electricity fairly decently.
It makes good resistors. Germanium is on the order of 10 to the minus 3. Silicon, anywhere from 10 to the minus 1 to 10 to the 1. So silicon is not a good conductor for the most part. It conducts a small amount of electricity.
Then you have insulators, which for the most part do not conduct electricity. So glass has a resistivity that's very high, 10 to the 9 to 10 to the 12. That's a lot higher than silicon. So for the most part, insulators, they just don't conduct electricity. Semiconductors, they don't conduct electricity.
they conduct a small amount of electricity. And metals, because they have such low resistivity values, they're very good conductors of electricity. Now, resistivity is a function of temperature.
There's an equation that you need to know. And here it is. PT is equal to PO times 1 plus alpha T minus TO.
So this means that the resistivity of a metal is affected by the temperature. Now, the situation is different for semiconductors. For metals, as you increase the temperature of the metal, the resistivity decreases.
I mean it increases, I take that back, not decreases, it increases. Now why is that the case? Well, whenever you raise the temperature of the metal, the free-moving electrons, they're colliding much more frequently.
And so, the drift velocity of the electrons caused by an electric current is much less because you have to go against all the random collisions that are occurring inside a metal and as we saw metals they have a much higher resistance at higher temperatures than at low temperatures so metals conduct electricity better at low temperatures rather than at high temperatures. If you cool a metal enough, like to a very, very cold temperature, a metal can become a superconductor, where there's virtually almost no resistance. So make sure you understand this. For metals, as you increase the temperature, the resistivity goes up. And so they have a higher resistance, which means that they're less conductive at high temperatures.
Metals conduct electricity better at low temperatures. low temperatures. Now semi metals or semiconductors like carbon, graphite, silicon, germanium, the situation is different.
As you increase the temperature they become more conductive they conduct electricity better and their resistivity decreases so it's different for semiconductors just keep that in mind metals they have a positive temperature coefficient so alpha is positive that's the temperature coefficient of resistivity and because alpha is positive for metals if you can go to your physics textbook and look at the table for resistivity You'll see that metals, they have a positive coefficient of temperature, which means that temperature and resistivity are directly related. If you increase the temperature, the resistivity go up. The semiconductors, they have a negative. temperature coefficient which indicates an inverse relationship as you increase the temperature the resistivity goes down which means that semiconductors are more conductive at high temperatures but metals are less conductive at high temperatures so why is this information useful why do we need to know the relationship between temperature and resistivity Well, sometimes you may want to digitally measure the temperature of an object.
And you can use a digital thermometer for that. And it's based on this principle. By knowing the resistance of a metal at a given temperature, and if you can measure the resistance at a new temperature, you can calculate that temperature based on that new resistance value. Now, r is equal to p times l over a. So, the ratio between, let's say, r2 over r1, that's going to be p l2 over l1.
I mean... P2 times L over A divided by P1 times L over A. I didn't put a subscript for L or A, assuming that the wire remains the same.
If it has a constant length and a constant area, and if only the temperature changes, then the resistivity will change only. So L and A will be the same, which means that they can be cancelled in this equation. So if we keep the length of the wire and the cross-sectional area the same, the resistance will depend on the resistivity.
And it makes sense. If you increase p, r will increase. So therefore, since this equation is pt over po, or p2 over p1, we can replace it with r2 over r1.
So we can say that rt is equal to ro times 1 plus alpha, t minus to. So, if we have the resistance at one temperature, and if we can measure the resistance at a new temperature, then using this equation, we can calculate the new temperature, or we can calculate the new resistance, whichever we want. And so that's how you can use resistance to measure the temperature of an object, or even the temperature of a room using a digital thermometer. Now let's work on some problems. A 15 meter long copper wire has a cross-sectional radius of 3 millimeters.
What is the resistance of the wire? And we're given the resistivity, which is 1.68 times 10 to the minus 8 at 20 degrees Celsius. And we also have the temperature coefficient, which is 0.0068. The equation that we need is this one, r is equal to p times l divided by a.
Now let's draw a picture. So let's say this is the wire, and l represents the length of the wire. And here is the radius of the cross-sectional circle that we see here, and the area of that is pi r squared.
So plugging everything into the formula, big R, which is the resistance, is equal to the resistivity, and that's 1.68 times 10 to the minus 8, multiplied by the length of the wire, which is 15 meters. divided by the cross sectional area, which is pi times the radius squared. Now the radius is 3 mm, and you need to convert that to meters.
To convert millimeters to meters, you can do this. divide by a thousand you can move the decimal point three units to the left or you can simply write three times 10 to the minus three just add 10 to negative three four millimeters because milli represents 10 to the minus three and don't forget to square So you should get .008913 ohms. If you're having issues getting this answer, make sure you put this in parentheses.
Otherwise, you're calculating my... divide by pi and then multiply by 3 times 10 to minus 3 squared. Part B, what is the resistance at 50 degrees Celsius?
Now what equation can we use to find a new resistance at a higher temperature? We can use this equation R is equal to The original R value multiplied by 1 plus alpha times T minus T naught, or the original temperature. So the resistance at 20 is 0.00. Alpha, the temperature coefficient of resistivity, that's 0.0068, that was given to us in the problem.
T is the new temperature, that's 50. T , the original temperature, which corresponds to this resistance, is 20. So what you want to do in order to get the answer, first subtract 50 and 20, which is 30, and then multiply 30 by .0068. You should get .204. Add 1 to it, and then multiply that by .008913. And this is equal to .01073.
Now, does this answer make sense? Well, the temperature was increased from 20 to 50. Anytime the temperature goes up, the resistivity of a metal will go up, so the resistance will increase. And that's what we see here. 0.01 is higher than 0.0089.
So as the temperature went up... the resistance of the copper metal went up as well. Now let's move on to Part C. If 200 milliamps of current flow through the wire, what is the voltage drop of the wire at 20 degrees Celsius?
The voltage drop is equal to the product of the current times the resistance. The current is 200 milliamps, but we need to convert that to amps. So you've got to divide it by 1,000. So that's going to be 0.2 amps, and the resistance at 20 is 0.008913. So the voltage drop...
is .0017826 volts. So this is the voltage drop per 15 meters, because that's the length of the wire. And because the number is so small, let's convert it to millivolts. Let's multiply by 1,000. So this is about 1.783 millivolts.
Now what about part D? What is the voltage drop per meter? So we need to divide this by 15. and you should get 0.119 mV per meter.
With this information, we can calculate the voltage drop at any length. So let's say if the copper wire was 300 meters long. All we need to do at this point is multiply it by 300 meters.
So these units will cancel and it's going to be point one one nine now. This is a rounded answer times 300 So the voltage drop is going to be thirty five point seven millivolts given a length of 300 meters So that's it for this problem. Number 2. A copper wire is connected to a 12-volt battery at 20 degrees Celsius and conducts a current of 0.45 amps. At a new temperature, the same wire conducts a current of 0.41 amps.
Is the new temperature higher or lower than 20? Well, the current decreased as the temperature changed, which means the copper wire is less conductive, and so its resistance is greater. The resistance of metals... will only increase if the temperature increases.
So therefore the new temperature is higher than 20 degrees Celsius. Now let's go ahead and calculate it. So first, we need to calculate the resistance at these two different temperatures.
So at 20 degrees Celsius, we can use the equation V is equal to IR, where R is basically V divided by I. So the voltage is 12 volts and the current at 20 degrees Celsius is 0.45 amps. So 12 divided by 0.45 will give us an R value of 26.667 ohms. Now let's calculate the R value at a new temperature.
So the voltage is still 12, but the new current is 0.41. 12 divided by 0.41 is 29.268. So that's the resistance at the new temperature.
The equation that we can use to find the new temperature is this equation. So the new R value is 29.268. R0, the original R value, is 26.667.
Alpha is 0.0068. 8 for the copper wire. We're looking for the new temperature T.
The original temperature that corresponds to this resistance is 20. So now, we just need to do math and solve for T. So the first thing we should do is divide both sides by 26.667. If we do that, we can get rid of the brackets that we have here.
So 29.268 divided by 26.667. That's 1.09754. And that's equal to this. Next, let's subtract both sides by 1. So, 0.09754 is equal to 0.0068 times t minus 20. After that, let's divide both sides by 0.0068. 0.09754 divided by 0.0068, that's equal to 14.34, and that's equal to T minus 20. So the last thing that we need to do is add 20 to both sides to isolate T.
So the new temperature is 20 plus 14.34, so it's 34.34. And that is the final answer. That's how you can calculate the new temperature given two different currents.