Hi there. So for this problem we have an in condensing light bulb that uses a coil of filament of to stand and we know that it has a length of 500 and 80 millimeters. Yeah. And we can write this in m zero 58 m now. Yeah, it has a diameter.
We also have that this has a diameter of 46 micro meters. When the temperature tour is gonna say to temperate tour Is of 20 C° And has a racist activity. We also have the resistive itty for the tips 10 and it is 5.25 times 10 to the minus aid arms parameter. Now this tell us that the temperature coefficient of racist activity, it's zero 0.0 0 45 Cells use degrees to the -1 and this remains. Okay.
So we are also told that the temperature of the filament increases linearly with current from 20 degrees two from 20° to 2000 520 associates degrees odd An occurring of one emperor. Okay, so the first question is the question A for this problem is What is the resistance of the light bulb at 20 Celsius degrees? When the temperature is 20 Celsius degrees? Yeah. So the first thing to identify is that this problem is about resistance of a light bulb filament and it can barriers with imperador. We know that the expression for the resistance at a given value of temperature is given by the project between the it's resistive. Itty times its length over the area.
It's gross exceptional area. So in this case we know the value for the resistive itty when The temperature is of 20 social degrees. Is this in here? So we just need to simply substitute that value. We also know the length of it which is 0.58 m. And we also can obtain the area because it will be the area for for um Yeah.
Mhm. For a slender in this case or um So we put all of those, all of those bodies in here. So there is a festivity has a value of 5.25 times. Then To the -8 arms per meter. The length of faith as a size set zero 58 meters.
And this over the area which is hi over the area of the car exceptional area of this filament. So we are given the um yeah, this in here is the diameter. So we obtain the radios of it. So we need to divide that value by two. So we will have And 46 times 10 to the -6.
Does me grow over to this to the square. So when we plot this into the calculator, we obtain that. The resistance has a value of 18.3 arms. So that's the racist activity of the filament when the temperature is of 20 soldiers degrees. Now, for part B of this problem, We are asked what is the current through the light bill? When the potential when the potential difference across east terminal is 120 bolts.
So party of this pope Party of this problem is when the difference, the potential difference is 120 bowls. So what we need to obtain is the current. When we have that value of the potential difference. So we need first to find the temperature or as a function of the current. We know that the graph of temperature versus The current is a straight line passing through zero at 20 Celsius degrees and one impair uh The value of this given volume here, 2000 and 520 social degrees.
So we can obtain the slope from that. And so this lobe is going to be 2500 C° per emperor. And there is a T inter step 20 Phil Hughes degrees. So using the slope intercept equation of a straight line, the equation for dad, the temperature or information of the current is going to be of the following. It's going to be That value 2500 C° and pair.
So with this multiplied by the current because we want that in function of the current And this plus 20 C degrees. Now we know that initial value. When we plot in here, the current equal to zero. We're going to attain 20C degrees. So we can obtain the resistance as a function of temperature.
Or so. The the formula for that is that the resistance in terms of the temperature or is given by the resistance are 20 Celsius degrees times one plus all from t minus the initial 20 Celsius degrees. This constant in here, we are given that value from the beginning. We know that the that this value is this zero point 0045. So we need to blow that value in there and we need to upload the value for the resistance at 20 and social degrees that we just obtained in the part a of this problem.
So we will have that the resistance at this temperature your is going to be so you know, So what what are recording to do now is to combine these two equations to combine these two equations, we plot the temperature in here. So what we are going to obtain is the following that the resistance in function now of the current because we are introducing one into the order is going to be this one plus alpha times um, that expression for that imperator, This is this times the current plus 20 C degrees. Now we note that the bolted or the potential difference is defined as the product between the resistance times the current. So that's what we are going to do. In this case we are plugging the value for the voltage and we are going to solve for the the current.
I so we will have the following will have this volume four. The potential difference, we have the value for There is a festivity at 20 C°. We have one plus we introduce all of the values that we know. 0.0 0 45. Um, seals use degrees over emperors.
There's times 2500 times the other. The current. We need to keep going with the other expression. So it will be 20 Celsius degrees. We close this in here and this in hot and this all of this is going to be times the current eyes.
So when we solve for I, we need to do some algebra in here and we we solve for the current. I we are going to obtain That the positive solution for this quadratic equation. We're going to attain a quadratic equation and the solutions are going to end. The solution for the quadratic equation is going to give us two values, one negative and one positive. Of course we need to take the positive solution and that positive solution is going to be 716 million pairs.
Um, so that's the value for the current. No. Um, the final question. Um, yes parsi of this problem. Yes.
What is the resistance when the potential is 120 bowls? Well this is parsi so. Oh no that's the one I just obtained. Oh no no no this is another when the potential is. Okay so that's part C. Of this problem.
So we want the resistive Itty. Okay the resistance. Sorry when the voltage is again 120 bolts and we just simply need we know that the resistance is the potential difference over the current. So we know these two values because the current is the one that we calculate in part B of this problem. So we just block that in there.
120 bowls over 716 million pairs. We know that million pairs is yeah it can be reading it, it can be reading us 700 sixteen's times 10 to the minus tree on purse. So when we plot this into the calculator, we obtain a value for the resistance of 167 arms. So that's the resistance. When we have that value for the potential um difference.
Now for part D. Of this problem, we are asked how much energy does the light bulb dissipate in one minute When 120 bolts is supplied across it's terminals. So again for this problem we have a potential difference of 120 bowls. But in this case It is applied in one minute. We know that one minute is 60 seconds.
Okay, so we want the energy dissipated during this one minute. So we used the following formula that the energy is equal to the power supplies times the time in that that is applied. So it it also is equal to um because the power can be expressed in terms of the current and the resistance. So it is the power can experience us. The current is four times the resistance and time is the time in this case.
So we are going to use our results from party and see that we just have obtained for the current. And we just need to just substitute those values in here. So we're going to obtain that. The energy is able to the current which is UB 716 million pairs. That is time during the -3 and purse is where the resistance of this um Of this filament of Houston is 167.
Well arms and the time apply is 60 seconds because that's one minute. So the energy that we obtain from this is going to be five point 14 kilo jewels. Now part E of this problem, ask how much energy does the light bulb dissipate in one minute when half of the talk of that voltage is supply. So we're now we have half of that voltage So we will have the report and then voltages 60 falters. If I'm correct.
Yes. And In this case again, the time is 60 seconds. So we want the energy when that we have that value for the potential difference. So we use that. The power in this case the power can also have the following expression.
Power is the product between the current times the potential difference. And we know all of these values. We get the current from the method that we obtain in part B With sits with 60 boulders. So in here We just substitute this by 60 bolt it. So when we solved that, we're going to obtain that, the current is going to be 0.40 nine 49 355 and purse mhm.
And the value of the value of the resistance. And we can obtain just by, oh, there is no space in here. So I'm gonna move this upward so we know that the resistance is going to be um the voltage over the current. So we know these policies. It's the boat over the current that we just have obtained.
So we attended the resistive itty, the resistance or the resistance is equal to 121.6 Emperors. So to obtain the energy, remember that it is the square of the at current times their resistance times the time of life. We know all of these values. We know that the The current 0.49, that value under from purse times these square times the resistance which is dispelled you. And the time that is 60 seconds.
So we obtained finally that the energy in this case is 1.78 kg jews. So that's it for this problem. Thank you..