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Copyright © 2009,2010 Joseph Mack
v20100223, released under GPL-v3.
Class lessons for a group of 9th graders who have taken my introductory programming course in python and who are working on the C programming course. This material was produced because the class didn't understand gates and registers and this was hampering their understanding of what the programs were doing. The section on mechanics has been moved to basic mechanics.
Material/images from this webpage may be used, as long as credit is given to the author, and the url of this webpage is included as a reference. The diagrams are produced by the GPL'ed package dia.
Table of Contents
Etymology of "Electricity" (http://en.wikipedia.org/wiki/Quantity_of_electricity).
The word electricity is derived from the Greek word electron for amber. When rubbed, amber developes a static charge (a deficiency or excess of electrons, depending on the material it is rubbed against). The charge is manifested by attracting other objects.
It took till the mid 1800's before anyone had much of an idea of how to use electricity (generate it, measure it), till 1897 before the electron was discovered by J. J. Thompson (http://en.wikipedia.org/wiki/J._J._Thomson) exploring cathode rays (the stream of electrons emitted from the cathode in the tubes used, till the mid 2000's, to display images in TV sets and computer displays), and till the mid 1900's before electrically driven consumer items became common.
For these lessons, electricity will be anything concerned with electrons free to move. Free electrons behave like a fluid e.g. water. Water and electrons both have pressure (voltage) and flow rate (current). It takes work to pump water up against gravity and it takes work to push electrons against and electric field. Work (energy) is released when water flows downhill (e.g. a water mill or hydropower station) and work is released when electrons flow with the electric field (e.g. in motors and light bulbs).
note: bring voltmeter
The water equivalent of voltage is force, pressure or height (water coming from an overhead tank). The unit is volts. The symbol for voltage (at least in the electrical engineering world) is E (for "electromotive force" How voltage, current, and resistance relate http://www.allaboutcircuits.com/vol_1/chpt_2/1.html). From what I can see, the people who learn about electricity, but have no intention of ever using this knowledge, use the symbol V, while the people who use their knowledge day in and day out, use the symbol E. (On the internet, the use of the symbol V is more common.)
The volt is named after the Italian physicist Alessandro Volta (http://en.wikipedia.org/wiki/Alessandro_Volta) (1745-1827) who invented the battery (pile) and used it to cause the muscles to twitch in freshly killed frog's legs. The definition of the volt (1946) comes from the ohm and the amp via Ohm's law (see below - it's the voltage across 1 ohm when a current of 1A is flowing through it). (see volt http://www.sizes.com/units/volt.htm).
Here's the voltages of common source of electricity. (for rechargeable batteries rechargeable batteries http://en.wikipedia.org/wiki/Rechargeable_battery; for energy required for visible light visible light http://www.windows.ucar.edu/tour/link=/physical_science/magnetism/em_visible_light.html)
source voltage lightning 100,000,000 high tension power lines 69,000 138,000 230,000 345,000 500,000 700,000 1,000,000 TGV 30,000 Shinkansen 25,000 urban train, earth moving equipt motors 10,000 (varies) urban power distribution 15000-10,000 household power (Europe, Australia, NZ) 240 household power (US, Canada, Japan) 117 lethal voltage somewhere in here (20-100) car battery (lead acid) 12 lithium battery 3.6 electron producing blue light 3.1 silver oxide battery 1.86 electron producing red light 1.8 dry cell (alkaline, LeClenche) batteries 1.5 NiCad 1.2 (no longer used, Cd is toxic) NiMH 1.2 solar cell 0.5 (note that this is less than the voltage needed to produce red,blue light)
That voltage is like pressure, can be seen in the production of a spark. If the gap is small enough (or the voltage is high enough), electrons will jump between two wires. The breakdown voltage of dry air is 33kV/cm (100kV/") (http://en.wikipedia.org/wiki/High_voltage). The spark in car ignitions is about 40kV. The gap is only 1mm or so, the higher voltage being required to jump across the fuel air mixture at high temperature and pressure.
When we talk about the voltage, we actually mean the potential difference between two points. There is no absolute measure of voltage. The same thing can be said about the force with which an object (say water) hits the ground; we can only tell the distance through which the object has dropped, not the absolute height at which it started (or stopped). If a point in a circuit is at 5V, we're saying that the point has a potential difference of 5V with an agreed upon location in the circuit, usually the ground represented by a line/wire across the bottom of the circuit diagram. In the lower half of this wiki circuit diagram (http://en.wikipedia.org/wiki/File:Circuit_diagram_%E2%80%93_pictorial_and_schematic.png), you would say that the resistor has (a potential difference of) 12V across it, or you could say that the voltage at the top of the resistor is 12V (the wire accross the bottom being accepted as the ground potential).
|Electricity can kill.
To get a shock you must make two contacts with the source of electrons:
one at the higher potential and one at the lower potential.
Birds sitting on a high tension power line, have their potential raised to the voltage of the power line,
but no current is flowing through them.
To get a shock (have current flow through them), the birds would have to put one foot on the ground.
Transmission line workers can safely be connected to EHV lines for inspection or repair.
This is called "live" (energized) or "bare hands" maintenance
(see bare hands EHV line inspection
(from the difficulty of finding this image,
I suspect that bare hands inspection of EHV and UHV lines is not common).
To kill yourself, depending on how you arrange things, you need only about 50V
(you can safely put one hand on each of the terminals of a 12V car battery
or button and regular batteries).
The usual immediate cause of death is stopping of the heart.
(People who survive a near lightning strike, usually have permanent damage to the nerves.)
The heart becomes involved if current flows through it and to do this,
current usually has to flow in one arm and out the other.
To prevent shock on systems below about 400V
If you're working on anything above about 50V, you will need safety lessons from someone familiar with those voltages and have them watch over you till you get the hang of it.
To make a battery:
put a drop of water on the paper, sprinkle a few crystals of salt on the paper (water is an insulator without the salt), put the paper between the two coins, measure the voltage: you should get 0.25-0.5V depending on the composition of the coins. Volta used Zn and Ag for his dissimilar metals. Using this method Volta was able to order metals into the electochemical series (http://en.wikipedia.org/wiki/Electrochemical_series). A metal higher on the list when paired with one lower on the list, would be +ve in a battery.
Current is the number of electrons passing a point/sec. The unit of current is the amp, being 1 coulomb/sec and the symbol for current is I (derived from the french for "intensity of current", but I can't find the URL). The water equivalent of electrical current is the flow rate of water (volume/time, or mass/time).
A coulomb is ≅6.24*1018 electrons. A typical AA battery will produce about 104 coulombs of electricity (electrons). (Compare a coulomb of electrons with a Mole = 6.03*1023 electrons. A mole of Na atoms (23g) and a mole of chlorine atoms (37.5g), together about the weight of a AA battery, when combined will produce 1 Mole=105 coulombs of electrons. A battery is not doing too badly in terms of efficiency.) The original definition of the "international ampere" (1893) was "that unvarying current that would deposit 0.001 118 000 grams of silver per second from a solution of silver nitrate in water" (history of the ampere http://www.sizes.com/units/ampHist.htm). With the atomic weight of Ag being 108, this is ≅10-5 moles of Ag.
The amp (http://en.wikipedia.org/wiki/Ampere) is named after Andre-Marie Ampere (http://en.wikipedia.org/wiki/Andr%C3%A9-Marie_Amp%C3%A8re) (1775-1836), who showed that that parallel wires carrying currents attract or repel each other, depending on whether currents are in the same (attraction) or in opposite directions (repulsion). This laid the foundation of electrodynamics. This experiment was used in 1948 to define the amp.
|I don't know how you can weigh anything to 7 significant figures. In the 1960s it was hard to weigh 1mg. Assuming you could weigh anything you like to 1mg, then to obtain a weight to 7 figures, it would have to weigh 11.18kg. To deposit this weight of Ag at 1A, would require 107secs≅115 days≅4 months. I can't imagine how anyone in 1893, much less today, could maintain a current of 1A stable to 7 figures for 4 months.|
Here's some typical currents
current source I,amps 300MW power station 1000 starter current (car) 100 household wiring 10-20 (1) 50W light bulb (110V) 0.5 watch battery 0.00002
|(1) 10A wire is stiff enough that short lengths (1m) don't bend under their own weight. 20A wire is stiff enough that it's hard to install.|
The unit of resistance is the ohm, and has the symbol Ω (from the Greek symbol omega, which presumably sounds like ohm). It is named after Georg Simon Ohm (1787-1854) history of the ohm (http://www.sizes.com/units/ohm.htm) who discovered Ohm's law.
The water equivalent of resistance is pressure drop/flow rate. To get more flow, you increase the water pressure, or increase the diameter of the pipe. Electrical resistance sets the amount of current flowing in a circuit, for any voltage.
The early definition of the ohm, the international ohm, was based on a physically reproducable object: a column of Hg.
The unit of electric resistance in the international system of electric and magnetic units established by the International Electrical Congress in Chicago in 1893. At first it was known as the reproducible ohm. The international ohm was defined as the resistance of a column of mercury of constant cross section at the temperature of melting ice, 106.3 centimeters long and with a mass of 14.4521 grams.
The ohm is now defined in terms of the quantuum Hall effect.
Values of resistance of descrete elements (resistors) in electronic circuits range from 1Ω to 10MΩ. Good connections, used to transfer power, have resistance in the mΩ range. Anything above about 20MΩ is regarded as an insulator (something that doesn't conduct electricity).
Ohm's Law: E = IR
Look at the lamp/battery circuit in How voltage, current, and resistance relate (http://www.allaboutcircuits.com/vol_1/chpt_2/1.html). Ohm's law says that if the battery is 12V and the current flowing is 4A that the resistance of the lamp is 12/4=3Ω
Power is measured in watts (http://en.wikipedia.org/wiki/Watt), with the symbol w. The unit is named after James Watt (http://en.wikipedia.org/wiki/James_Watt), who improved the Newcomen steam engine, producing the condensing steam engine. This engine powered the Industrial Revolution which gave millions of unemployed children, no matter how young or inexperienced, the opportunity of gainful employement 12hrs a day (http://www.theonion.com/content/news/industrial_revolution_provides).
Power is work/time (how fast you do work). A watt is a Joule/sec.
The power delivered to (or produced by) an electrical device is given by
P = E*I
Example: a lightbulb powered by 110V uses 50W. How much current is flowing through it  ?
While power produced is just fine, in electronic circuits, when power is dissipated in a device, it gets hot. The problems with heat are
As integrated circuits get smaller, they must use corresponding less power to stay at the same temperature (because they have a smaller surface to radiate the heat).
Here's the power levels associated with familiar objects/activities (some of these I saw on a URL I can't find again).
activity power electrical power station 10-1000MW train locomotive 5-10MW car engine (highway) 100kW radio, TV station 50W-1MW kitchen oven 1-5kW athlete 200-500W (1) household lighting 40-500W light from sun 1kW/m^2 power to maintain body temperature 50-100W (2) audio power, radio,TV 1-10W (speakers are about 1% effecient, i.e. output is 10-100mW) human voice 1-10mW signal at radio, TV receiver 1nW-1uW light from 3.5mag star 1nW/m^2
|(1) I've ridden a stationary bicycle powering a 200W TV set. I could only keep it running for about a minute.|
|(2) Most of our food consumption goes to maintain our body temperature, rather than physical activity. The resting rate is given at about 60w (the amount clothing is not specified) doubling with activity basal metabolic rate (http://en.wikipedia.org/wiki/Basal_metabolic_rate). 2000calories/day is 100W.|
A reformulation of the P=EI equation above, using Ohm's Law (E=IR) is
P = EI E = IR P = I^2R
If you want to transfer power from one place to another you can do it by increasing the current or the voltage (or both). If you increase the current, then you need thicker cables to reduce the resistance or an increasing voltage drop will occur along the line. This equation says that power dissipated in a cable increases with the square of the current. As well, as you increase the current, the voltage drop along the cable is proportional to the current (i.e you won't be delivering the same voltage at the other end of the cable). As the current is increased, thicker cables will be needed to increase the power handling capability. Thicker cables are expensive (you need to buy more copper, you need to support it and you need to install it). The problems with increasing voltage are it's lethality (but once you're above about 100V, you're just as dead from 100kV as from 100V) and insulation (wires can't be allowed to short to a chassis, other electronics, or people). It turns out that insulating high voltage wires is cheap and simple (often you can hang them in free air), as long as you handle the lethality by only letting trained people service them. People have decided that if you want to transmit power, you do it at the highest voltage (and lowest current) practical.
The heat capacity of air at 0°C is 1.297kJ/(m3.C) (i.e. it takes 1.297kJ to heat 1m3 of air by 1°C) (see specific heat capacity http://en.wikipedia.org/wiki/Specific_heat_capacity). How long would it take a 129.7W lightbulb to heat 1m3 of air at 0degC through 1°C  ?
Here we talk about powering cars with electricity.
Cars in most of the world run on gasoline (petrol). The ideal engine for a car would have
In this article ( hydraulic power units http://www.hydraulicspneumatics.com/200/TechZone/ManifoldsHICs/Article/True/6405/TechZone-ManifoldsHICs) Fig 3 shows that the electric motor has the required torque and speed characteristics.
Fig 4 shows the torque and power curve for a gasoline (petrol) engine. The gasoline engine
See diesel-electric (http://wikicars.org/en/Diesel-Electric_Hybrid). A better arrangement is to have a petrol/gasoline engine (efficiency 25-30%) or even better a diesel engine (efficiency 40%) (see engine efficiency http://en.wikipedia.org/wiki/Engine_efficiency) working at constant rpm driving an (electric) generator, which in turn drives electric motors on the wheels. (The efficiency of standard automotive generator/motors is about 60%, rising to 95% for generators in hydropower stations. Apparently higher efficiency is possible 96% efficiency for a 100W motor (http://techon.nikkeibp.co.jp/english/NEWS_EN/20090403/168295/).) The diesel engine would operate at constant speed no matter what the car was doing, and no transmission (or differential) is needed.
This scheme is in use right now with diesel trains and large (mining grade) earth moving vehicles
and the Tourneau L-1800 for which I couldn't find a photo.
Let's say we were to adopt this system for cars of power 100HP (=74.8Kw; let's round that up to 100kW) what current and voltages would have to be used to transmit power from the generator in the engine bay to the motors on the wheels?
100Kw = 100V * 1000A; voltages OK, thick cables = 1000V * 100A; lethal voltages, cables still thick = 10000V * 10A; lethal voltages, managable cables (household thickness, but with better electrical insulation)
The thing to do then is do what trains and heavy equipment operators are doing; change to diesel electric cars. Why isn't it being done. I could imagine that no-one wants to deal with the lethal voltages. People got used to driving cars with highly inflammable fuel and no-one worries about that anymore. You would only want the vehicles serviced by people who knew what they were doing so you could start with buses. Diesel-electric buses are being produced by New Flyer Industries, Gillig, Orion Bus Industries, and North American Bus Industries.
If you have a garden hose and you pump more water in at one end (i.e. increase the pressure), then water is available to come out the other end. How fast does this water become available? You could imagine attatching a piston to the far end of the hose and hitting the piston with a hammer; how long would it take for the pressure pulse to propagate to the other end of the hose? The answer is the speed of sound in water i.e. 1484m/s at 20°C (about 4 times faster than in air - see speed of sound http://en.wikipedia.org/wiki/Speed_of_sound).
This assumes rigid walls for the pipe. When I was younger, you could turn off a tap (faucet) (in the house) suddenly and the moving column of water in the house piping would slam into the valve. The rise of pressure would propagate back through the piping at the speed of sound making a big CLANK from all the pipes in the house. Pipes then were iron or copper. I can't get my house pipes to do this today. Possibly it's because they're all PVC and flex, absorbing the pulse, rather than reflecting it back through the house piping.
In the body's circulation system, the heart provides pressure pulses to pump the blood through the system. The capillaries that transistion between the arterial and venous flow are fragile and will rupture under high pressure. A rupturing of the capillaries in the brain is a medical emergency and can lead to a stroke or death. If the walls of the pipe flex or are elastic (like the arteries in your circulation system), then pressure pulses will be absorbed in the walls, slowing down the velocity of and reducing amplitude of the pulse. The elasticity of arteries decreases with age and by hardening of the arteries caused by, among other things, deposition of plaques of cholesterol. A demonstration at a local museum (the Durham Museum of Life and Science http://www.ncmls.org/) used to have a display of two pulsating (piston) water pumps. One had the water flowing through a rigid transparent tube about 1m long; there the water came out in spurts. The other had the water flowing through a piece of tubing the same size, made of silicon rubber. You could see the silicon rubber expand and contract with each pulse; there the water came out in a continuous stream.
When we look at the water coming out of the hose, we see that it's moving much slower than the velocity of sound - it's moving at walking or running pace. From this we see that there are two velocities in a fluid
Let's look at the same problem for electrons going through a wire. If the voltage is changed at one end of the wire, the new potential is propagated along the wire at about 50-95% of the speed of light. However the electrons pumped in at one end of the wire travel quite slowly. Let's see how slowly. The calculation here SPEED OF "ELECTRICITY" 1996 Bill Beaty (http://www.eskimo.com/~billb/miscon/speed.html) gives 8.4cm/hr. This speed is called the "drift speed" of electrons.
|The calculations at Bill's site get the actual numbers. The principle is simple; say you have a wire 10cm long, with 10 electrons in it and you push an electron in one end, then another electron will come out the other end (presumably not the one you pushed in). Each electron in the wire occupies 1cm and the electrons move 1cm each time a new electron is pushed into the wire. If you increased the diameter of the wire by 10^0.5, then the cross section (and volume) of the wire would increase by a factor of 10. There would now be 100 electrons in the wire and adding an extra electron at one end would now only move the electrons in the wire by 0.1cm - i.e. at 1/10th the speed. The drift speed of electrons then depends on the ratio of free electrons in the wire, compared to the number of electrons being transported by the electric current, and on the cross sectional area of the wire. It turns out that the number of free electrons is enormous compared to the number of electrons transported by any current we can push down the wire.|
The drift speed depends on the current and the diameter (or cross section) of the wire. (for more about drift speed, see Microscopic View of Ohm's Law http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html). The reason the drift velocity is so small compared to the speed of light is that there's a lot of electrons in a wire and only a small fraction of them are needed to supply the electricity coming out the end of the wire. Bill Beatty (this para) calculates that 1 coulomb of free electrons in Cu occupy a cube of side 0.4mm, about a grain of sand. In 30 gauge wire (about 0.4mm diam), 1A current would move 0.4mm/sec.
Here we use dimensions to compare the mechanical and electrical analogues (force,EMF), (mass flow, current), (mechanical resistance=pressure drop/flow rate, electrical resistance). While this approach is reassuring in the case of the first two pairs of quantities, the relationship between mechanical and electrical resistance is not so obvious (sometimes understanding comes slowly and with difficulty).
With the list of available dimensions, what is the dimensions of the volt  ? I've been saying that voltage is a bit like force. Let's see if we can find a relationship between electromotive force (E), whose units are volts and force, whose units are newtons.
dimensions of force F = m.l.t^-2 dimensions of E E = m.l^2.t^-2.c^-1 conversion between E and F E = F.(l/c) definition of work/energy W = F.l so E = W/c
Voltage is work per unit charge, i.e. the amount of work you have to do for each electron to move it against an electric field (or the work recovered allowing the electron to move with the electric field). What's the water equivalent of W/c (i.e. voltage)? Let's try work/mass. What is work/mass?
W = F.l W/m = F/m.l but F/m = a so W/m = a.l = g.h #going with (or against) gravity.
The mechanical equivalent of voltage is W/m, which in the direction of the earth's gravity is height*(accel of gravity). Voltage is a little like the height you have to lift a mass through (but not exactly, because with height, you need the dimension of acceleration too).
What's the dimensions of the ohm  ? This is quite complicated. The mechanical resistance to flow of water in a pipe is (pressure drop)/(flow rate). Let's see if we can relate electrical resistance to mechanical resistance in a pipe. We could measure mass flow rate (mass/sec, dimensions = m.t-1) or volume flow rate (volume/sec, dimensions = l3.t-1). Let's use mass flow rate (we can convert later to vol/sec using density, which has units of mass/volume and dimensions m.l-3).
Let's devise a formula for the (pressure drop)/(mass flow rate): If we increase the cross sectional area of the pipe, we expect the flow to go up; if we increase the length of the pip, we expect the flow to go down. Let's assume that the flow rate is proportional to the area and inversely proportional to length. Because we're only interested in dimensions, assume all constants are 1.
flow rate (mass/sec) = pressure*area/length check for dimensional correctness Pressure = F/area dimensions pressure = (m.l.t^-2)/(l^2) = m.l^-1.t^-2 LHS RHS m.t^-1 (m.l^-1.t^-2).(l^2)/l = m.t^-2
These dimensions don't match. To make them match, we need to multiply one side (or other) by a fudge factor (what you want/what you've got). Do we have any justification for doing this? We know that the flow rate depends on the fluid being pumped through the pipe; the velocity of water and peanut butter will be quite different under the same conditions.
|In computational fluid dynamics, peanut butter is the standard check in thought experiments, to see if your code is working or if your results are sensible. You can be modelling water flow around the pylons of a bridge, air over a curved surface and your first check has to be "what would happen if my fluid was peanut butter?"|
Yes there we have it! There's a property of fluids that we haven't allowed for - how well they flow. Let's call this new property viscosity and throw in there all the dimensions we're missing. (When accountants do this it's called "creative accounting"; when scientists do this it's called "an exciting discovery".) Which side of the equation do we want to put viscosity on; the left or right? Let's say we decide that increasing viscosity reduces flow, then viscosity goes on the LHS. What's the dimensions of viscosity?
viscosity(unit?).flow rate (mass/sec) = pressure*area/length viscosity = (m.t^-2)/(m.t^-1) = t^-1
Viscosity has the dimensions t^-1. What does this mean? Quantities with dimension t^-1 are called (among other things) relaxation (or decay) phenomena. If the quantity is disturbed in some way, it returns to the original state at a rate determined by a rate constant (dimension = t-1) e.g.
are all governed by the same equation. The decay law is exponential and the rate of decay is governed by the time constant.
fraction remaining = e^-(t/t0)
where t0 is the rate constant.
Example: The half life of the fissionable nucleide 235U (http://en.wikipedia.org/wiki/Uranium-235) is 7*109years. What is the rate constant for the fission of 235U
|the natural log (log to base e, called ln()) can be found with bc by echo "l(number)" | bc -l|
The rate constant is the time for the perturbation to relax back to e-1=0.37 of its original perturbation. Engineers use the rate (or time) constant all the time, since the equations are much simpler. Why does the public use the 1/2 life (time constant/0.69)? Well why does the public use Fahrenheit, feet, and lbs, buy tickets in lotteries and accept a society without universal health care? You tell me. You would have to tell the public that their calculations would be simpler if they'd use the number which gives 0.37 of the original amount of material. I worked as a post-doc at the University of Maryland doing chemical kinetics. Much to the consternation of my supervisor, I calculated all my numbers as rate constants. He had to convert them to half quantities before he could grasp their meaning. If tenured professors at universities have trouble using rate constants, then we have at least another generation before we can expect the public to use them.
What's the significance of the dimension(s) of viscosity being t-1? If you make a small perturbation in your fluid (stick a pencil into your peanut butter), the definition of viscosity we're using here is the time taken for the perturbation to return to 0.37 of its original displacement (there are other definitions of viscosity; in gases a more convenient one is the rate of transport of momentum).
In case you've forgotten we're finding the relationship between electrical and mechanical resistance. So what's the dimension of physical resistance in a pipe (pressure drop)/(flow rate) now that we've discovered viscosity?
(pressure)/(flow rate) = viscosity*(length/area) = (t^-1)*l^-1 = l^-1.t^-1 = (lt)^-1
What this represents physically is not obvious to me. Let's compare physical resistance to electrical resistance.
R = m.l^2.t^-1.c^-2 = physical_resistance.(m.l^2.t^-1.c^-2)/(lt)^-1 = phys_resistance.m.l^3.c^-2
Again, this was not terribly enlightening.
While it's easy enough in mechanical systems to grasp the dimensions of the quantities velocity, work and power, in electrical systems, the dimensions are more complicated. Instead you check that volts/amps/power (and simple combinations) match on both sides of equations.
E = IR 110 = I*220 I = 0.5A
E = IR 12 = 6*R R = 2ohms
P = E*I 50 = 110 * I I = 0.5A
1m^3 of air needs 1.297kJ to heat it 1degC a 129.7W lightbulb is using 129.7J/s. It would take 1.297*1000/129.7=10secs to warm the air by 1deg
P = EI listing dimensions LHS RHS m.l^2.t^-3 E * (c.t^-1) thus E = m.l^2.t^-2.c^-1
E = IR listing dimensions LHS RHS m.l^2.t^-2.c^-1 (c.t^-1).R dimensions of R R = m.l^2.t^-1.c^-2
fraction remaining = e^-(t/t0) 0.5 = e^-(7*10^9/t0) ln(0.5) = -7*10^9/t0 t0 = -7*10^9/ln(0.5) yrs = 10^10 yrs
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