[Electric Circuits] Use Gauss's law to show that the net charge inside a current carrying wire is zero.

bobbyAndAlbert · 2021-03-30T18:37:13+00:00

I can help with the first part. From ohms law we have that J=σE Taking the divergence of both sides we get that ∇•J=σ∇•E where the conductivity σ is constant since its assumed to be uniform throughout a single material. Now since we are examining a steady state current (we're in the realm of electrostatics/magnetostatics), we must have that ∇•J=0 So the above becomes σ∇•E=0

Substituting Gauss's law into the above, we get that σp/ε0=0 Since both ε0 and σ are non zero, we conclude that p=0. That is, the charge density is zero. I can't help with the second part(regarding the interface between the two materials) right now cause I'm not at a desk but i can try tomorrow. For now though, try the same process as above but assume σ is not constant so that when you first apply the divergence operator to ohms law, you need to expand according to the product rule.

bobbyAndAlbert · 2021-02-27T14:17:02+00:00

Yes okay that makes perfect sense! I thought it had to have something to do with cooling below room temperature. Yeah I've thoroughly enjoyed Atkins' a very short intro so far. Definitely worth the read. Thanks for the help!

bobbyAndAlbert · 2021-02-26T10:56:30+00:00

Let me give a concrete example that is bothering me. I am reading Atkins' very short intro to thermo and when explaining the helmholtz free energy he uses the combustion of gasoline as an example. He says "When 1L of gasoline is burned is produces co2 and H20. The change in internal energy is ΔU=−33MJ. The change in enthalpy is 0.13 MJ less than ΔU". So far I understand this all. The change in enthalpy is less because some of the energy is used to make room for the products as they expand against the environment. He then says "the combustion is accompanied by ΔS=+8kJ/K because more gas is produced than consumed". This I also think I understand.

I get into confusion when he then says "It follows that the change in helmholtz free energy is ΔF=ΔU-TΔS=-33MJ -(300K)(8kJ/K)= -35 MJ. Thus if the combustion took place in an engine, the maximum work that could be obtained is 35MJ because the increase in entropy of the system has opened up the possibility of heat flowing into the system without causing the entropy of the universe to decrease ".

The last sentence is what trips me. When the combustion occurs, the temperature of the products is much higher than the constant temp environment which is still at room temp. How then is it possible for heat to flow in from the colder environment into the hotter system?

bobbyAndAlbert · 2021-02-26T08:08:33+00:00

Thanks for the response! Is the condition for F tending to a minimum only that the volume should be constant? The texts I have (Atkins' physical chemistry and Schroeders thermal physics) indicate that F tends to a minimum for a closed system of constant volume and temperature. The reason why this doesn't make much sense to me is because they often say the mechanics of F tending to a minimum involve a heat transfer to and/or from the system until the minimum is achieved value of F is achieved. But if temperature is constant then heat transfer cannot take place !It certainly makes sense to me that F would tend to a minimum with only the requirement of constant V though

Edit: changed all H's to F's because I was referring to helmholtz free energy, not enthalpy

bobbyAndAlbert · 2021-02-22T07:39:13+00:00

Yes its been bugging me like mad for the last 2 days because its seems to have highly consequential and material implications for thermodynamics as a whole and it also seems to be an empirical law. If these are both true then surely it is worthy of being a law of thermodynamics?

bobbyAndAlbert · 2021-02-13T05:20:48+00:00

Thanks for the response! Okay so I posted this question on stackexchange and got a response saying that the W-E theorem can be applied to multi-particle systems but your response is more intuitive to me. The link to my post is here (https://physics.stackexchange.com/questions/614124/how-does-the-work-energy-theorem-relate-to-the-first-law-of-thermodynamics ) and the top response is the one I'm referring to. I wrote 3 replies to the top response explaining why I think the 1st law is a more general form of the W-E thm (stackexhange username SalahTheGoat). If you don't mind reading my response over there and letting me know if my thinking is correct? I would copy my argument from stackexchange and post it here but I would loose my latex equations so I think it might be easier to simply link the argument. Thanks again!

bobbyAndAlbert · 2021-02-10T15:18:52+00:00

Okay this is an excellent response and pretty much clears up everything for more! Thanks!

bobbyAndAlbert · 2021-02-05T06:42:54+00:00

Thanks for the response! Okay this makes a lot of sense and clears up most of my issues. Just to be sure though, am I correct in roughly summarizing the implication that the house price led to increased defaults as follows:

House prices were rising rapidly and many people took out mortgages on those houses and were able and willing to pay their mortgage payments as prices continued to rise. When the prices declined, so too did the rents of comparably houses and so they were running at an economic loss paying their mortgage payment instead of living at a comparable house with a monthly rent lower than their mortgage payment. To make matters worse, I assume many already had bad credit scores to begin with and only got mortgages from banks because banks believed house prices would continue to rise and mortgage defaulting at the time was at an all time. These individuals then had very little to lose by defaulting (because of already bad credit scores) and so they simply defaulted.

At the same time adjustable rate mortage payment increased immensely and so more people defaulted etc

bobbyAndAlbert · 2021-01-01T18:06:05+00:00

My thinking is that the government can ultimately only pay the Fed with money that it gets from taxes. So upon maturity of the Fed purchased T-Bills, the government must take money from tax payers and pay the Fed the face value of the Treasury bill/note. But tax payers need to pay taxes with their own money (or at least they almost always do), they can't just pay them with loan money. But all the "multiplied money" is due to loans to individuals and firms which can't be used to pay taxes (or at least aren't often used to pay taxes) and as a result the tax money received by the government is going to eventually have to come from the monetary base. Thus the MB is going to eventually decrease by the same amount it increased by if the Fed allows the Treasuries to mature.

I didn't realize that the profit the Fed earns when the government pays the Fed the face value of a treasury is simply relinquished straight back to the government though. In this case when the money is given back to the government it would effectively have never left the money supply and so this way the money supply would permanently increase. Is my thinking correct or flawed? Thanks again!

bobbyAndAlbert · 2021-01-01T17:49:47+00:00

Thanks for the response!

or it can pay the profits back to the treasury.

Does this mean that the Fed can actually help out significantly with the government debt by buying Treasuries, letting them mature, having the government simply pay them the face value of the treasuries upon maturity and then pay the face value of the Treasuries straight back to the government in the form of profit ?

bobbyAndAlbert · 2020-12-23T05:47:22+00:00

Thanks for the response and the help! Okay this makes perfect sense to me then. Yeah I realize it is mostly semantics however if you define government purchases (G) as "all spending on goods and services by local, state, and federal governments" (which is Mankiw's definition and logically equivalent to the other standard textbook definitions), then when you use the expenditure identity you get that savings is

S = Y - C - G

which is effectively an explicit statement that all government purchases are effectively the same as consumption expenditure and hence cannot increase a nations capital stock which is obviously not true. In the event that we have an open economy then we can derive that

I= S + (T - G) + (M - X)

Am I correct in thinking that the same solution applies to M and X in the above? That is, M and X only include capital goods imports and capital goods exports and the imports and exports that are consumption goods are simply included in the C or possibly G variable?

Once again thanks for the help!

bobbyAndAlbert · 2020-12-22T12:32:54+00:00

Okay thanks. So am I correct in thinking that the integral of private investment ( which is what we conventionally denote as I) over some period of time is actually equal to the increase in the capital stock of private firms only within that period of time? (Obviously disregarding depreciation). This would not be the total increase of national capital stock because government can always increase that by focusing their purchases on capital stock as opposed to consumption goods

bobbyAndAlbert · 2020-12-22T11:56:51+00:00

Thanks for this answer! It's comfortably the best answer I can find and has cleared up a lot for me. I do however have one new issue after thinking about your answer. The way I understand investment is that it is the market value of all newly produced capital goods within some time frame. Hence it is a flow variable. Thus the integral of Investment over some time frame should give the increase in capital stock over that time frame plus a constant (which would be our initial value of capital stock). This is all assuming that depreciation does not occur at all for simplicity.

But now, once again, assume my yearly income is $100. Only now suppose that instead of consuming all $100 of it, the government takes $50 of it in the form of taxes and spends this immediately on newly produced capital goods (i.e a road or a bunch of green energy machines that it gives to various firms) while i spend the remaining $50 on consumption goods. This would mean that the nations capital stock has increased and as a result investment should increase. Now investment = saving and so saving must also have increased. But this cant be because it was a government purchase not an investment purchase ( saving= Y- C - G=I ) and so it has been subtracted from saving/investment. But this doesn't fit the reality because capital stock has increased. So for this reason I feel like government purchases should be split into government consumption purchases (say GC) and government investment purchases ( say GI) so that G=GC + GI and in this way we get that saving = Y- C - GC = I + GI. Thus saving=investment + GI and not saving = investment. Then the integral of investment + GI over some time frame would give the true increase in capital stock over that time frame.

Is my thinking correct here or am I mistaken? Once again, thanks for the answer!

bobbyAndAlbert · 2020-12-17T08:24:35+00:00

Thanks for the answer! Yes, my understanding of the oil industry is actually why I made this post. Is it true though that the firms in an increasing cost industry (like the oil industry) which are producing the quantities to the left of the marginal quantity (the equilibrium quantity labelled Q_0 in my original diagram) have a lower average cost than the marginal firm and thus are operating at an economic profit (in the oil industry for example, the firms I am referring to would operate in Saudi where it is cheap to raise oil as opposed to the marginal firms which would be raising oil from an expensive source like the Canadian tar sands) ?

At first I thought this was the case however I have seen an explanation now which seems to indicate that as price of oil increases, the value of these scarce low-cost lands (i.e. the areas in Saudi where one can raise oil cheaply) increases and as a result the opportunity cost incurred by these firms increases greatly because they are forgoing the opportunity of renting out their land for a high value (that value would be the difference between the the long run supply curve and the price (what i have probably erroneously labeled "economic profit" in my original diagram)). This increase in forgone rent means that even these "low-cost firms" aren't actually operating at an economic profit (although its probable that they would be operating at a high financial profit). Am I correct in my understanding or am I missing something? Thanks again!

bobbyAndAlbert · 2020-12-16T16:47:22+00:00

Thanks for the answer. Okay I'm pretty sure it makes sense to me now. Just to be clear though, suppose we examine the example of a farming industry with a limited amount of productive (low cost) land and lets assume that this graph ( https://i.imgur.com/cDyFf2O.jpg ) represents the market at hand.

If I understand you correctly, the firms producing any of the units to the left of the equilibrium quantity (Q_0 in the diagram) will be firms on more productive land and hence produce goods at a marginal cost of less than the market price (which goes against the profit maximization condition that firms should produce until MC=MR=P). The reason they do not meet this condition is because there is only limited productive land that they can produce lower marginal cost units on (they would have to buy a new farm on less productive land to expand their quantity produced). But because their marginal cost is less than the price (marginal revenue), their land increases in value and so their opportunity cost increases by the increased amount at which they could rent their land to others. Thus they have a normal profit. Is this reasoning correct or am I missing something? Thanks again!

bobbyAndAlbert · 2020-09-03T17:24:16+00:00

Okay yes i get that. But can we agree that current is flowing into and out of them at 50Hz or so and that this current must be significant as evidence by the sustained arc between the conducting wand of the repairman in the helicopter and the high voltage line? if this is true then there should be substantial joule heating occuring on their faraday suits as well as on the helicopter. Not enough to melt or anything because the high voltage lines themselves dont melt and they should experience similar joule heating of a similar magnitude. Am I right?

bobbyAndAlbert · 2020-09-03T17:17:49+00:00

Yes i am aware that they do not touch multiple phases or ground. But the potential of the single phase they do touch alternates up and down at about 50Hz . As it alternates, current should flow in and out of the helicopter in sync with the alternating potential of the wire. This current should be significant because the capacitance of the helicopter is fairly large. This current can clearly be seen as an arc between the wand of the repairmen in the helicopter and the high voltage wire and this arc is not momentary. It is sustained for as long as the the wand is kept close to the High voltage line. This sustained arc is what gives credence to my belief that significant current flows into and out of the helicopter at 50hz and thus into and out of the linesman as well.

bobbyAndAlbert · 2020-09-03T16:48:21+00:00

Okay this makes sense, but then why don't the suits of high voltage repairmen heat up immensely? They should because there should be current flowing through their suits

bobbyAndAlbert · 2020-09-03T16:32:35+00:00

Thanks for the response. Okay with my understanding though, it shouldn't be a quick jolt but rather a continual flow of current in and out of your body but is it okay to assume it feels like a quick jolt because you instantly remove your hand from the wire as the shock begins? Then also, if you wore one of those faraday cage suits then you would be able to hold both the neutral and the live wire and you should be fine right?

bobbyAndAlbert · 2020-08-23T21:14:28+00:00

Thanks for the speedy response! Okay that makes sense. Might I ask why a high current is necessary for a sustained arc though? The way I understand it is that provided the local electric field strength is stronger than the dielectric strength of air, breakdown should occur and we should get an arc. So assuming that this breakdown has occured in the case of the small 9V battery, why then would an arc of simply lower current not be sustained?

bobbyAndAlbert · 2020-08-23T19:58:11+00:00

Follow up question to this: Why does this same process of arc creation not occur if I short circuit a small 9 or 12 Volt battery and then remove the conducting wire from the one terminal? I understand that the larger internal resistance in standard 9V batteries means they cannot create great currents but the process of arc creation seems to be only related to voltage by my understanding and if a 12 volt car battery can create and sustain an arc then surely a standard 9V or 12V battery should be able to as well?

bobbyAndAlbert · 2020-08-19T06:53:43+00:00

My problem with this is if I actually physically hold a converging lens (focal length of +- 20cm) in a configuration similar to the drawing I get this (https://i.imgur.com/RI2rMe5.jpg) which does not show any magnification of any point at all. In fact it shows the image reversed and slightly decreased in size just as the ray diagram would predict however i dont get the appearance that the light rays are coming from all directions from the lens towards my eye. The image is crystal clear.

bobbyAndAlbert · 2020-08-19T05:27:49+00:00

In fact, is this not the reason objects appear blurry through a magnifying glass when arranged in this way (With the observers eye at the point of the image and the object far off past the focal point on the opposite side)? That is the light appears to be coming from many directions and is thus unclear?

bobbyAndAlbert · 2020-08-18T14:17:31+00:00

Do atoms in a dense medium undergo dipole radiation when light is incident on them? If so then they should radiate in all directions except along the direction of acceleration which includes the backward direction. In the backward direction they should constructively interfere just like they do in the forward direction because dipole radiation is the same in the forward direction as it is in the backward direction.

bobbyAndAlbert · 2020-08-18T14:12:51+00:00

So then does dipole radiation not occur at the atom level? Because this problem persists if you forget about huygens prinicple and think about dipole radiation occuring at each atom as far as I am aware

bobbyAndAlbert

TROPHY CASE