In the bridge circuit shown below, find the value of the resistance \(c\) such that the equivalent resistance connected across the voltage source is also equal to \(c\). Show that for this value of \(c\), the voltage across \(c\) is equal to \((\sqrt{b}-\sqrt{a})/(\sqrt{b}+\sqrt{a})\). Try to solve this problem yourself before looking at the solution below.
Start by finding the equivalent resistance. The easiest way to do this is to replace the voltage source with a current source \(I\), and then find the voltage \(V_1\) at node 1. The equivalent resistance is then \(V_1/I\). This involves solving 3 equations for nodes 1, 2 and 3. The equations are Kirchoff's current law applied to the nodes. Leaving out the details, the equivalent resistance is
\[R = \frac{2ab+(a+b)c}{a+b+2c}\]
If you set \(R=c\) and solve for \(c\), you get \(c=\sqrt{ab}\). In other words, when \(c\) is equal to the geometric mean of \(a\) and \(b\), the equivalent resistance is equal to \(c\).
To solve the second part of the problem we need an expression for the voltage across \(c\). Going back to the voltage source and writing the node equations for nodes 2 and 3, we get
\[(1/a + 1/b + 1/c)V_2 - (1/c)V_3 = (1/a)V_1\]
\[-(1/c)V_2 + (1/a + 1/b + 1/c)V_3 = (1/b)V_1\]
Solving these 2 equations for \(V_2/V_1\) and \(V_3/V_1\) we get the following expression for the voltage across resistor \(c\).
\[\frac{V_2-V_3}{V_1} = \frac{(b-a)c}{(b+a)c+2ab}\]
If you substitute the value \(c=\sqrt{ab}\) into this equation, you find that
\[\frac{V_2-V_3}{V_1} = \frac{\sqrt{b}-\sqrt{a}}{\sqrt{b}+\sqrt{a}}\]
This simple circuit can calculate the ratio of the sum and difference of the square roots of 2 numbers.
Sometimes you have to measure a voltage that has a high output impedance. To do this accurately you need to know the input impedance of your voltmeter. Ideally it should be much much larger than the impedance of what you're trying to measure. The following figure illustrates the situation.
The voltage we're trying to measure is \(V_x\) and it has an output resistance of \(R_x\). The meter has an input resistance of \(R_m\). The voltage measured by the meter will then be
\[V = \frac{V_xR_m}{R_x+R_m} = \frac{V_x}{1+R_x/R_m}\]
So you can see that \(V\approx V_x\) only when \(R_m>>R_x\). If you happen to know the values of \(R_x\) and \(R_m\) then you can determine what \(V_x\) is even when the two resistances are of the same magnitude. How do you measure the input resistance of your voltmeter? If you apply a known \(V_x\) using a known \(R_x\) then you can solve for \(R_m\) in the above equation.
\[R_m = \frac{R_x}{\frac{V_x}{V}-1}\]
We did this for the old Scope multimeter you see in the following picture using \(V_x=12v\), \(R_x=1M\Omega\).
The meter read \(10.98v\) which gave us \(R_m=10M\Omega\). This is pretty typical for most handheld multimeters.
Some really cheap multimeters such as the one shown below that we got at Harbor Freight Tools showed an input resistance of only \(1M\Omega\). You have to be careful about what you measure with a meter like that. In some cases it can give you very inaccurate readings.
"The Refrigerator and the Universe: Understanding the Laws of Energy" by Martin Goldstein and Inge F. Goldstein is gold. The authors know the subject of thermodynamics very well and take us on an epic journey through the far reaches of the subject's domain, as well as cover its history. While this book doesn't contain problems, it is an inspiration for problems and further investigations. I couldn't recommend it more highly. My only criticism is really just a personal preference that the book be at the level of a university physics textbook, with calculus, not just algebra. But as such, it appeals to a wider audience. The extensive annotated bibliography at the end serves as a roadmap for further reading.
Things do look a bit bleak right now but human beings can sometimes perform miracles, the COVID vaccines and the James Webb telescope are proof of that. Nothing is ever perfect and a new year inspires new ideas for making the world a better place and life a little more pleasant. So here are some of our ideas.
Let's start with something simple like the calendar. Why are there 7 days in a week when we have 365 days in a year? If you divide 365 by 7 you get 52 with one day left over, we have 52 weeks plus one day in a year. This means dates do not fall on the same day of the week every year. New Year's Eve this year is on Friday but next year it will be on Saturday. How do we keep it on Friday every year?
The solution is simple, we change the number of days in a week to 5 instead of 7. We can safely get rid of Monday and Tuesday since nobody really likes those days anyway. Saturday and Sunday will still be the weekend i.e. days where you can do any damn thing you want. The work week will only be three days: Wednesday, Thursday, and Friday. There will be 73 weeks in a year. That's 73 weekends, surely we can all get behind this!
The one possible objection to this new scheme is the fact that a year is actually about 365.25 days long. So we need leap days or we'll eventually be celebrating Christmas in shorts and tank tops the way the Australians do. But we don't need to make leap days a part of the regular calendar. We'll just call them free days, a special holiday where politicians have to clean public toilets on live TV and homeless people get to sleep and eat in the White House.
Now let's look at clocks. There are 24 hours in a day so why not use a 24 hour clock. Forget this AM and PM baloney. 1 PM is simply 13:00, 6 PM is 18:00, and so on. Isn't that simpler? If there's 24 hours then number them that way.
But the most confusing thing about clocks is that I may know the time where I am but I have no idea what the time is on Pitcairn Island. We need one universal time for the whole world. Yes that means some people will be getting up at 7:00 while others will be going to bed at that time but everyone will get used to it. Also, if you go to some other country you don't have to do any calculations to figure out what your people back home are doing right now.
Another more radical solution is to do away with clocks altogether and replace them with a device that shows the longitude where solar noon is occurring at any given moment. This provides a universal time and it can tell you if it's night or day time at any given place if you know it's approximate longitude.
One of the great insanities of this world is the belief that we need economic growth at all costs. If GDP goes down it's seen as some kind of great catastrophe, the Federal Reserve starts printing money like crazy, making bankers ever richer, politicians are voted out of office, and businesses are urged to start hiring and ramp up production. Does it matter what's produced? No, anything will do. A million new bobble head dolls is great as long as it adds to GDP. Most of this new economic "growth" is just crap that will end up in a landfill in a few years. This kind of attitude will eventually destroy this beautiful planet we live on. It also consigns so many people to boring, meaningless lives which is probably the biggest tragedy. We can change the world by buying less crap and making more beautiful things.
Stand on a street corner in any American town of a few thousand people or more and watch the cars go by. The vast majority of them will have a single occupant. Very few will be small and fuel efficient, or better yet, electric. A significant number of them will be monstrously large pickup trucks pumping out clouds of black diesel exhaust. Do we really need giant machines that weigh tons to haul one human being maybe a mile or so down the road? Why can't more people use bikes for short trips around town or to get to work? At least in cities, we should give the same priority to bicycle transportation as we do to automobile transportation.
If you haven't tried bike riding recently, you're missing out on lots of fun. There's something exhilarating about moving along under your own power outside in the fresh air. It's good exercise and will help with that New Year's resolution to lose some weight. It also gives you a whole new perspective on the world. You will see things you never noticed before. Give it a try.
Do you like to save money? Do you like getting something for free? If so then get yourself a clothes line and stop using your electric or gas powered clothes dryer. It doesn't take much to put up a clothes line in your backyard. It's the simplest way to take advantage of free solar energy. Your clothes will smell great and they will last longer, plus you get more fresh air and sunshine. And since most electric power in the US is produced with fossil fuels, you're likely to be adding less CO2 to the atmosphere.
There's good evidence that a plant-based diet can prevent many of the diseases that cause so much human suffering. Cancer, diabetes, and heart disease can be prevented and reversed in people who already have them. It reduces not only human suffering but the suffering of animals too. The industrialized farming of animals is a horrifying thing to see. The conditions under which these animals are forced to live out their short lives before they end up on somebody's dinner plate is a moral and ethical transgression of the first order. There are videos of animals being funneled into a slaughterhouse. You can see the absolute terror on their faces when they get to the point where they finally realize what's going on. Please, let's stop this insanity.
We'd love to see more people have jobs that they enjoy, reducing traffic accidents caused by the mad rush to get home after work to have a little enjoyment at the end of the day.
And last but not least, we'd love to see kindness and compassion increase among humanity.
We wish everyone a happy, healthy and safe New Year.
© 2010-2023 Stefan Hollos and Richard Hollos