For some applications, it is important that the value of a res...

Question

For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a 3.60-k Ω resistor from a carbon resistor and a Nichrome wire-wound resistor connected together so the total resistance is the sum of their separate resistances. What value should each of these resistors have (at 0°C) so that the combination is temperature independent?

Accepted Answer

Welcome back. Everyone. In this problem. A 4.5 kg ohm heating element is made from graphite and platinum resistors connected in series determine the resistance values for both resistors at zero °C to ensure the total resistance remains constant with temperature changes. A hint is that the temperature coefficient for graphite is negative 0.0008 per degree Celsius. And for platinum is 0.00392 per degree Celsius. A says the resistance for our graphite is 3.7 kg ohms while for platinum is 750 ohms B says it's 3.7 kg ohms and 760 ohms respectively. C 3.8 kg ohms and 760 ohms respectively and D 3.8 kg ohms and 770 ohms respectively. Now, what are we trying to figure out here? Well, we want the resistance value for both resistors. OK. For our problem that tells us the total resistance value because we know that our heating element has a resistance of 4.5 kg ohms. Now, how can we relate that total resistance to help us find the resistance for our graphite resistor and the platinum resistor. Well, recall that the resistance is dependent on the temperature and to find that resistance depending on the temperature is going to be equal to the value of the resistance. In this case, are not multiplied by one plus alpha T where alpha represents the temperature coefficient and T represents the temperature. So in this case, that means then for our graphite, OK, or, or graphite, we'll have a resistance where R not equals RG. OK. So the resistance four or graphite equals RG multiplied by one plus alpha GT. OK. And likewise, it's going to be different for platinum. For platinum, it would be RP OK, multiplied by one plus alpha PT. Now we know that the total resistance since both of them are connected in series is going to be equal to the sum of our resistance. So in other words, OK, in other words here, let me uh get rid of this. In other words, this tells us then that our total resistance are total at any temperature T can be written as RG multiplied by one plus alpha GTK plus RP multiplied by one plus alpha PT where RG is the resistance of the graphite resistor and RP is the resistance of the platinum resistor at zero °C. No notice here, we are told that we want to find the resistant values to ensure the total resistance remains constant with temperature changes. So that means we're looking for the resistance values in the condition of temperature independence. And for the resistance to be independent of temperature, the temperature dependent terms should cancel out. In other words, in that scenario, that means IG alpha G plus RP alpha P must be equal to zero. Now, if we can solve for either RG or RP, in terms of the other resistor, then we should be able to substitute that to solve for output our resistance. Now let's go ahead and solve a RP. We know that our temperature coefficient for graphite, OK is negative 0.0008 per degree Celsius. And we also know that our temperature coefficient for our platinum resistor is 0.00392 per degree Celsius. So we know that both of those will be equal to zero. That means then we can go ahead and isolate RP. We can add this expression for our graphite resistor to both sides and then divide by the constant to get RP to be equal to RG multiplied by 0.0008. OK. Divided by 0.00392. Know that we have an expression for our platinum resistor. In terms of our graphite resistor, we can solve for our total resistance because from the total resistance in this condition, that is the condition for our temperature independence. That means our total resistance equals the value for our graphite resistor plus our platinum resistor. And we already know the total resistance is 4500 ohms because it was a 4.5 kg ohm heating element. So that means 4500 is going to be equal to RG plus or expression for RP in terms of RG, which is 0.0008 RG divided by 0.00392. Now when we add both of these terms, OK, then we should find that it equals 1.2041 RG. Therefore RG equals 4500 divided by 1.2041 which equals 3737.23 ohms or approximately 3.7 kg ohms to two significant figures. Since our total resistance, OK? Is our total resistance is 4500 ohms. That must mean then OK, that must mean then that using our value for our graphite resistor in our expression for platinum resistor RP is going to be equal to 0.0008 multiplied by 3737.23 ohms divided by 0.00392 which equals 762.7 ohms. OK? Or approximately 760 ohms to two significant figures. Therefore, our graphite resistor is 3.7 kg ohms while our platinum resistor is 760 ohms. If we look back on our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Temperature Coefficient of Resistance

Series Resistors

Resistance Matching

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