Monday, February 22, 2021

Notes to a video lecture on

Relativity - Transformation of Space-Time Coordinates
(notes to item #3 of Einstein's "Electrodynamics")

The following is an example of how a system of linear equations can be used to derive formulas of special theory of relativity. Albert Einstein has derived these formulas in his "Electrodynamics" in a more physical, more intuitive way. The following is pure mathematics and, as such, causes much less problems in understanding.

1. Assume we have two systems of coordinates, one stationary with coordinates {X,T} (assuming for simplicity all the movements will occur in one space dimension along X-axis and one time dimension) and another with coordinates {x,t} moving along the X-axis with constant speed V.
2. Assume that at time T=0 systems coincide (i.e. X=0, t=0 and x=0).
3. Assume that the speed of something, as measured in both stationary and moving systems is the same and equal to C regardless of the direction of the movement (that "something" is the light in vacuum, but it's physical characteristics are unimportant)
4. Assume further that we are looking for linear orthogonal (i.e. preserving the distance between points and angles between vectors) transformation of coordinates from (X,T) to (x,t) that satisfies the above criteria. What would this transformation be?

Linear transformation from {X,T} system to {x,t} system should look like this:
x = pX + qT
t = rX + sT
where p, q, r and s are 4 unknown coefficients of transformation, which we are going to determine by constructing a system of 4 linear equations with them.

We should not add any constants into above transformations since {X=0,T=0} should transform into {x=0,t=0}.

A. Notice that the property of orthogonality is needed to preserve geometry (i.e. no deformation) and, therefore, to preserve the form of all physical equations of motion. As it is well known, orthogonal transformations have determinant of the matrix of coefficients equal to 1, i.e.
ps − qr = 1. An unfamiliar with this property student can study this subject separately (we at Unizor plan to include this into corresponding topic on vectors).
The above is the first equation to determine unknown coefficients. It's not linear, it is rather quadratic, but the rest of the equations will be linear and that's why we included this particular problem in the topic dedicated to linear systems.
We need three more equations to determine all the unknown coefficients.

B. Since moving system moves along X-axis with speed V, its beginning of coordinate (point x=0) must at the moment of time T be on a distance VT from the beginning of coordinates of a stationary system. Hence, if X=VT, x=0 for any T. From this and the first transformation equation x = pX + qT we derive:
0 = pVT + qT or
0 = (pV + q)T.

Since this equality is true for any T,
pV + q = 0
and, unconditionally, q = −pV.
This is the second equation for our unknown coefficients.

C. Since the speed of light C is the same in both systems {X,T} and {x,t}, an equation of its motion in the stationary system must be X = CT and in the moving system x = Ct. Therefore, if X=CT, then x=Ct. Put X=CT into equations of transformation of coordinates. We get x = pCT + qT, t = rCT + sT. Substitute these expressions into x=Ct:
pCT + qT = rC2T + sCT.
Reduce by T,
pC + q = rC2 + sC.
This is the third equation for unknown coefficients.

D. Repeat the logic of a previous paragraph for the light moving in the opposite direction with a speed −C. We get, if X = −CT, then x = −Ct. Therefore, x = −pCT + qT, t = −rCT + sT and (since x = −Ct)
−pCT + qT = rC2T − sCT.
Reduce by T,
−pC + q = rC2 − sC.
This the fourth equation for unknown coefficients.

So, this is the system of equations for 4 unknown coefficients of transformation p, q, r, s:
(a) ps − qr = 1
(b) q = −pV
(c) pC + q = rC2 + sC
(d) −pC + q = rC2 − sC

It's not exactly linear, but it has sufficient number of linear equations (all but one) to solve it using the known methodology. Let's solve this system of equations by combining the methods of substitution and elimination. We will express unknown variables q, r, s in terms of p using equations (b), (c) and (d). Then we will substitute them into (a) to get an equation for p. Solving it will allow to evaluate all other unknowns.

E. From (c) and (d), adding and subtracting these equations, we get:
2q = 2rC2, therefore q = rC2
2pC = 2sC, therefore p = s

Since from (b) q = −pV,
−pV = rC2 and r = −pV/C2.

Now all coefficients of a transformation are expressed in terms of one unknown coefficient p. To get the value of p, use the first equation (a).

F. Substituting q, r and s, expressed in terms of p, into an equation (a) ps − qr = 1, we get:
p2 − (−pV)·(−pV)/C2 = 1, therefore
p2·(1 − V2/C2) = 1 and
p = 1/√1−V2/C2 .

From this all other coefficients of a transformation matrix are derived:
q = −V/√1−V2/C2 
r = −(V/C2)/√
s = 1/√

G. The final transformation matrix looks exactly like in Einstein's article on electrodynamics, but seems to be much simpler to arrive at and the derivation is strictly mathematical.
x = (1/√1−V2/C2 )·X − (V/√1−V2/C2 )·T
t = ((−V/C2)/√
1−V2/C2 )·X + (1/√1−V2/C2 )·T

Traditionally, factor V/C is replaced with Greek letter β, which results in formulas:
x = (1/√1−β2 )·X + (−V/√1−β2 )·T
t = ((−V/C2)/√
1−β2 )·X + (1/√1−β2 )·T

One more simplification is usually done by introducing Lorentz factor γ equals to 1/√1−β2 :
x = γX − γVT
   = γ(X − VT)

t = −γVX/C2 + γT
   = γ(T − VX/C2)

The final form of transformation of coordinates in the Special Theory of Relativity is:
x = (X − VT)/√1−(V/C)2 
t = (T − VX/C2)/√1−(V/C)2 

Monday, February 8, 2021

Electronics - Diode: UNIZOR.COM - Physics4Teens - Electromagnetism - Usage

Notes to a video lecture on

Electronics: Diode

To the category of electronic devices we relate all devices that use electricity and whose primary purpose is not just to heat or to do mechanical work.
Radio, television, computers, all kinds of non-mechanical switches and regulators, phones, hardware that runs Internet and many others devices belong to this category of electronic devices.

Obviously, we cannot talk about how all these devices work. Instead, we will spend some time to understand the basic components that are used in these devices.

Diodes are one of the simplest electronic components that are present in practically all electronic devices.
The main purpose of a diode is to let the electric current only in one direction. This process of allowing electric current to go only in one direction is called rectifying the current.

The origin of diodes lies in an observations made by scientists and engineers (Fleming, Edison and others) at the end of 19th century. Two wires that did not touch each other were placed inside a vacuum tube, one connected to a positive pole of a battery (anode) and another connected to a negative one (cathode). At normal temperature there was no electric current between them, because they did not touch each other. But, if the negative wire was heated, some electric current between these wires was observed, while heating the positive wire did not cause any current in the circuit.

The explanation of this phenomenon is simple. Heating increases the activity of elementary particles inside a negatively charged metal of a wire that has an excess of electrons. With this increased activity certain electrons escape from the surface of the metal and form some kind of electron cloud. This is called thermionic emission.
In the presence of a positively charged wire some electrons from a cloud are attracted to a positive wire, thus forming a current. New electrons from a negative poll of a battery replace the escaped electrons, maintaining a fresh supply of new electrons, which enables a steady current.

Obviously, if a positive wire with its deficiency of electrons is heated, even if some electrons do escape because of high temperature, they will be repelled by a negative wire and attracted back to a positive one. No current would be formed.

Below is a schematic representation of a vacuum tube diode with ampere meter in a circuit showing the existence of electric current.

The symbol for a diode in electronic schemas is

The primary usage of diodes is to rectify the alternating current, where they allow the current to go only in one direction, thus converting AC to DC.
They are also used in signal isolation, filtering and mixing.
Vacuum tube diodes are used now only in high capacity rectifiers with semiconductors based diodes used in all the electronic devices we usually deal with.

Let's analyze the process of rectifying AC using diodes.
As we know, the current in a regular AC circuit is sinusoidal, changing the direction and the value.
If a diode is included into a circuit in sequence, the current in one direction will go through, while it will be prohibited to go in the opposite direction.
This causes the alternating current to change from a regular sinusoidal wave-like behavior into irregular direct current.

The irregularity of the current in a circuit can be improved by using a bridge rectifier built from 4 diodes as follows

If the AC generator produces positive charge at point A and negative at point B, the flow of electrons is in the direction BEFNMDCA. The electric current is defined as going against the flow of electrons in the opposite direction ACDMNFEB.

If the AC generator produces positive charge at point B and negative at point A, the flow of electrons is in the direction ACFNMDEB. The electric current is defined as going against the flow of electrons in the opposite direction BEDMNFCA.

As you see, the direction of a current in both cases is from point M to point N, regardless of polarity of a generator contacts.

The AC current rectified by diodes that form a bridge is better than rectified by a single diode, but still is quite irregular, comparing with DC current from a battery.
Additional improvements can be achieved by splitting a current into two separate lines and putting a capacitor on one of them to change the phase of the oscillations and then combining signals together.
Here is how the combination works.

The combined Signal 1+2 still has some irregularity, but is much more stable than each of its components Signal 1 or Signal 2.

In general, by combining currents, shifted by phase relative to each other, helps improving the stability of the flow of electrons. Real life rectifiers are build on this principle.

Vacuum tube diodes have been largely replaced by semiconductors, but the principle of their work is very similar.

Thursday, February 4, 2021

Electric Devices: UNIZOR.COM - Physics4Teens - Electromagnetism

Notes to a video lecture on

Electric Devices

In this chapter we will discuss different usages of electricity by grouping them according to certain criteria.

Two major groups to differentiate are electric and electronic devices.
To the category of electric devices we relate devices that use electricity for two primary purposes - to produce mechanical work, like rotation, and to produce heat, like in electric stove, including producing light by means of heating, like in incandescent lamps.

Regardless of such a simple definition of this category, the number of devices in this group is enormous, and these devices are the first ones invented to make our lives easier. Examples of these electric devices are a subject of this lecture.

To the category of electronic devices we relate all other devices that use electricity and whose primary purpose is not just to heat or to do mechanical work. They will be discussed in the next lecture.

Mechanical Work

The easiest and most common example of the usage of electricity to produce mechanical work is an electric motor. This is a device that converts electricity into rotation.

For alternating current numerous one-phase and three-phase motors are used all around us.
The primary motion they produce is rotation, which in some cases is converted into other forms of motion.

They pump water, rotate fans, work in refrigerators, rotate wheels of electric trains, lift elevator cabins, drill for oil and gas, operate machinery at manufacturing plants, move construction cranes.

Direct current in most cases comes from batteries and is used in direct current electric motors, like the one that starts the car engine, rotates a hard disk in a computer, rotates the battery powered electric drill etc.

Electric wall clock is another example of using electricity to move the wheels of a clock. Some electric clocks work off alternating current, some off direct one.

Washing machine has an electric pump to deal with water pumped in and out and another motor to rotate the drum.

Just as an illustration, let's calculate the technical characteristics of a motor that should supply water to a building, where I live in.

The water is pumped to the roof tank, then it flows down to all apartments.

We have 12 floors, each about 3 meters high, 200 apartments, each apartment needs about 100 liters of water during 3 hours in the morning.
So, the pump should pump 200·100=20,000 liters of water to the height 12·3=36 meters during 3 hours time.

This allows us to calculate work W performed during this time and power P of the pump needed to perform this work.

Each liter of water has a mass of 1 kg and, therefore, the weight of 9.8 N.
W = 9.8·20,000·36 =
= 7,056,000 J

Since the time to do this work is T = 3 hours and each hour has 3,600 seconds, the power of the motor is
≅ 653 J/sec

Usually, we need some excess of power to prevent shortage during some extra work requirements and to account for losses of power in the motor itself due to friction and heating, so a motor of about 1000 watt (1 kilowatt) should suffice, if we allow it to work without interruption.

In practice, the motor should start and stop periodically, depending on the level of water in the tank, so it has to pump faster than water is consumed and we need a more powerful motor, say 1.5 KW.
With voltage to such a motor at the level of 220V the current flowing through this motor is
I = 1500W/220V ≅ 6.8A

In addition, considering that things break and we need an uninterrupted water supply, we need the same pump with the motor of the same power to be ready to automatically pick up the load in case the main pump breaks.
That makes a design a bit more complicated with two pumps working in parallel, alternating their work and, in case one breaks, another working alone. This requires some electronic switching mechanism.


Electric heater and incandescent lamp represent this group of electric devices. They warm and light up our homes.

We use electric stove to prepare our food.

Electric hair drying fan is an example of a combined mechanical (to push the air) and heating (to heat up the air) electric device.

Drying machine uses electricity to rotate a drum, produce heat and blow it into the drum using a fan.

For illustration, let's do some calculations related to incandescent lamps.
Consider a lamp with marked power consumption of P=100W and voltage U=120V.
Here we are talking about alternating current and, therefore, all characteristics are effective.

The effective electric current running through it is
I = 100W/120V ≅ 0.8333A
The resistance of the spiral in this lamp is
R = 120V/0.8333A = 144Ω

Obviously, we can check that
P = U²/R = I²·R
With given voltage in the circuit, the power consumed by an incandescent lamp will be more when the resistance is less. That's why a lamp consuming 100W has a thicker spiral (with less resistance) than a lamp consuming 40W for the same voltage.

Another interesting example of using electricity to produce heat is welding. This is a process when an electric arc between two electrodes is formed and used to melt metal.

There are many different types of welding machines. An important characteristic is the electric current going through the arc formed between electrodes. Usually it's in hundreds of amperes, like 500A-1000A with voltage in the range 30V-60V.
That makes the power consumption of a welding machine to be somewhere from 15KW to 60KW, which is a lot, comparing to a power of about 1.5KW needed for a water pump described above.

These characteristics fluctuate as the welding process goes, they depend on the length of an electric arc and materials used as electrodes.

Friday, January 29, 2021

Electricity to Consumers: UNIZOR.COM - Physics4Teens - Electromagnetism ...

Notes to a video lecture on

Distribution to Consumers

When we described the grid in the previous lecture, we concentrated on main principles of its work - synchronization of the generators of electric energy with the grid. It might make a wrong impression that the grid is one gigantic wire that have certain voltage, frequency and phase, and all the generators must adhere to this standard, when connected to it.
The real situation is more complex.

Considering the grid covers large distances, we must maintain the high voltage in it to avoid waste of energy to heat (hundreds of thousands of volts). Most consumers, however, need relatively lower voltage (no more than a few hundred volts).

To connect each consumer to a grid through a powerful and very expensive transformer is impractical. Instead, we combine a group of geographically close consumers into it's own grid, connected at one point to a main ultra high voltage grid through a transformer that lowers the voltage, and then we connect consumers to this second level grid.

For example, the whole city can form this secondary grid with lower voltage than in the main ultra high voltage grid. Thus formed, this secondary grid can go to each street with lower voltage better suitable for consumption. And, because this grid is relatively localized, there will be no big loss of energy to distribute electricity at a lower voltage.

Making the picture even more complex, we can arrange third level grid lowering the voltage for each building in the city from street level of the secondary grid to a lower voltage building level that goes to each apartment.

Yet another complication can be introduced by connecting other generators to a grid. Since our grid now consists of many grids connected via transformers from main ultra high voltage to second level with lower voltage in the city and third level for each building, we can introduce generators into each grid, and the only requirement for this is to make sure that each generator conforms to a corresponding grid's voltage, frequency and phase.

For example, a building's management decided to put solar panels on the roof. They generate electricity for a building and, if there is excess of energy, it goes to a grid for general consumption. That, probably, is the third level grid that serves this building, it has it's own characteristics, and the output of solar panels must conform to these characteristics.

Similarly, a city decided to build a power plant working on burning the garbage. This power plant produces electricity that should go to a grid that serves the whole city, which we call the second level grid. The voltage produced by this plant is higher than the one in any building level grid but not as high as in main ultra high voltage grid that supplies the whole city.

The overall picture of distribution of electricity consists of different grids of different voltage, connected through transformers, having producers and consumers in each grid. Each consumer of electricity should have its parameters to be the same as the grid it's connected to (voltage and frequency). Each producer of electricity should be synchronized with the grid it's connected to in voltage, frequency and phase.

On the picture below we have schematically displayed the generation of electricity at 13,000V, transformers that increase the voltage to 600,000V to transmit along the long distances, transformers that decrease the voltage to 7,200V at the entrance to a city that supplies this electricity to buildings and, finally, transformers that decrease the voltage to 240V before entering buildings.
Inside the buildings this voltage is distributed to individual apartments.

In practice there are more devices participating in the grid, stabilizing the voltage, sensing the abnormal conditions, controlling different functions of the grid, protecting the grid against disastrous conditions, attacks or human errors etc.

The grid is constantly changing as new sources of energy come on line, new consumers are attached to a grid, new more efficient maintenance devices are introduced

Sunday, January 24, 2021

Electric Grid: UNIZOR.COM - Physics4Teens - Electromagnetism - Distribution

Notes to a video lecture on

The Grid

The grid is an extremely important solution to most problems with electric power interruption due to different technical issues at power plants that generate electricity.

Consider a simple analogy of distributing water to apartments in a large building from a tank on a roof.
If you have one tank, and it needs repair or cleaning, the water to all apartments must be shut off while the work performed.
If, however, you have two water tanks on the roof both connected to a common distribution pipe, from which the water is flowing to all apartments, we can just close the connection of the tank to be cleaned to a distribution pipe, leaving another tank operational and water supply to apartments uninterrupted.

This same principle is used in combining many generating electricity power plants to a common distribution wiring, thus assuring uninterrupted power supply. This system of interconnected power plants forms a grid that feeds those consumers of electricity connected to it, assuring their uninterrupted work.

Obviously, with electricity it's much more complex than with water supply.

Let's enumerate problem we have to resolve when connecting different electric generators to a common power distribution system.

Direct Current

Let's connect two batteries generating direct current parallel to each other and power up a lamp.

The proper connection requires the same polarity (positive pole of one battery is connected to positive pole of another, negative - to negative) and the same voltage generated by these batteries. Only then there will be no electric current between the batteries, only from a battery to a device consuming the electricity (a lamp in this case).
These two conditions, similar polarity and equal voltage, are necessary and sufficient conditions to successfully connect two batteries in parallel. The overall energy capacity of these two batteries will be twice as big. They will last twice as long on the same load (one lamp) or they can double the load (have two lamps parallel to each other) and serve the same time as one battery on a single load.

Alternating Current

Analogously, two generators of alternating current (AC) must have the same output voltage, if connected in parallel. Otherwise, there will be an unnecessary electric current between them, which diminishes their usefulness.

But for AC generators there are more characteristic parameters than just a voltage. Voltage varies as a sinusoid with time and is characterized by amplitude, frequency and phase.

All three parameters must be the same for a proper parallel connection of two generators. Their output voltages, as functions of time, must coincide exactly to each other. And this is a big challenge to build a grid with many different generators, each contributing their part in overall power supply.

The above considerations dictate strict restrictions on how any new source of electricity, like a new power plant or a new solar panel are hooked to a grid.

First of all, the output of a generator at the point of connection to a grid must be alternating. Solar panels, for example, produce direct current, so, before connecting to a grid, the DC electricity must be converted to AC. Special devices called inverters provide this type of conversion, assuring the frequency of voltage oscillation to be that of the grid.

Then, depending on a point of connection to a grid, the output voltage must be equalized with that of the grid at the connection point. This can be done with proper transformers.

Finally, we have to adjust the phase to synchronize the output of the generator with the phase of a grid. This can be achieve by adjusting the speed of a generator's rotor while monitoring the phase difference on a special sensor until proper synchronization is achieved.

Overall, the connection to a grid is a sophisticated process that requires special devices, tools, instrumentation and care.
There are many controls that monitor, adjust and maintain the regime of work of a grid. In many ways it's automated, computer controlled and reliable. However, human errors do happen and some of them result in significant distortions of power supply to large areas and affecting a lot of people. As an example, the 2003 blackout in Ohio resulted in power loss across Eastern United States and even some areas in Canada.

Friday, January 22, 2021

Electricity In Transit: UNIZOR.COM - Physics4Teens - Electromagnetism - ...

Notes to a video lecture on

Electricity in Transit

Let's discuss how electricity is delivered from the power plants to consumers.

The only way to deliver the electricity from the place it's generated to a place it's used is via electric wires.
Since the distance between the power plant and a consumer can be substantial, may be even hundreds of kilometers, the problem of losses of electric energy in transit because of wire resistance is extremely important.

Examine a simple electric circuit consisting of a source of electricity (generator) and a consumer (like an electric motor).
In theory, we have four places where electric energy is spent:
(a) inside a generator due to internal resistance,
(b) inside a wire from a generator to a motor due to wire resistance,
(c) inside a motor due to useful work the electricity does and internal resistance,
(d) inside a wire from a motor back to a generator due to wire resistance.

Obviously, only in a motor the energy is spent with some useful purpose, in other places the energy is spent just because it's unavoidable, that is wasted.

Amount of energy wasted to heat per unit of time due to wire resistance Wwire depends on the current running through wire Iwire and the wire resistance Rwire according to a formula
Wwire = I2wire·Rwire

For practical example, let's calculate the amount of energy wasted in some long piece of copper wire.
The resistance of a wire depends on resistivity of material it's made of ρ, is proportional to the length of a wire L and is inversely proportional to its cross-section area A
Rwire = ρ·L/A

For copper the resistivity is approximately
ρ≅1.70·10−8 Ω·m.
Assume, the combined wire length to and from a consumer of electricity is
L = 1 km = 1000 m
and its diameter is
D = 2 mm = 2·10−3 m,
which gives the area of its cross-section
A = π·D²/4 ≅ 3.14·10−6
Then the resistance of this peace of wire is
Rwire ≅ 5.4 Ω

For example, we are supposed to run an electric motor working at voltage
Umot=220 volt
and delivering power of
Wmot=2.2 kilowatt.

Then the current it requires is
Imot = Wmot/Umot = 10 A

The current Imot=10 A must go through the wire
Iwire = Imot = 10 A
Then the amount of energy wasted to heat in the copper wire of resistance Rwire=5.4 Ω per unit of time (a second) is
Wwire = I²wire·Rwire = 540 W

For a price of about $0.1 per kilowatt this amounts to about $0.054 per second. For 24 hours uninterrupted work the financial waste amounts to $4,665, and that is every day of operation of one motor, which is absolutely unacceptable.

Reducing the resistance of a wire by making it thicker or using multiple parallel wires has its practical limitations because of cost of wires. Therefore, our solution to reduce the energy wasted to heat due to wire resistance, while staying within reasonable limits with the cost of a wire, must be related to reducing the current Iwire running through a wire without reduction of power that is supposed to be delivered to consumers of electricity.

This can be accomplished by using transformers.
Immediately after generation, the alternating current is directed to a transformer that increases the voltage and proportionally decreases the amperage.

At the output of this transformer the voltage reaches thousands of volts - from low voltage of 1000V to ultra high voltage above 800,000V, depending on the length of wires from generators to consumers.
This high voltage electricity is delivered to consumers, where another transformers reduce the voltage to standard needed to run all their different devices.

Consumers of electricity get the voltage required to run their equipment, but the current running in the long wires between generators and consumers is low, thus reducing waste of electric energy.

Consider an example above with a motor that needs Wmot=2.2 kilowatt of electricity at voltage Umot=220 volt and, therefore, requires Imot=10 A electric current.
If, instead of transmitting electricity with these parameters, we increase the voltage by a transformer before sending it to long wires to, say, 2200V, thus proportionally reducing the amperage by the same factor, our amperage will be
A = W/U = 2200/2200 = 1A
Reducing the amperage from 10A to 1A reduces the energy waste by a factor of 100 because the heat formula depends on a square of amperage.

The distribution of electricity, therefore, should include transformers that increase the voltage before sending electricity along long wires and decrease it wherever it's needed for usage by consumers.
With this modification the picture that corresponds to practical aspects of distribution of electricity looks like this

To increase the electrical systems' reliability, improve the energy balancing and make sure of uninterrupted power supply, the sources of electrical energy (electric power plants and other installations producing electric energy) are combined into a network called the grid.
The principles of this networking are a subject of the next lecture.

Sunday, January 10, 2021

Electricity at Power Plants: UNIZOR.COM - Physics4Teens - Electromagneti...

Notes to a video lecture on

Electricity at Power Plants

Addressing distribution of electricity, we will primarily discuss the way electricity, produced at the power plants, is delivered to consumers.
We will concentrate on the process of distribution of electricity generated from the kinetic energy of rotating turbines, as the most quantitatively significant source of electricity.

Three stages of this distribution are
(a) at the power plant
(b) in transit
(c) at consumers.

This lecture is about what's going on at the power plant that produces the electricity from the kinetic energy of rotating turbines.

Turbines at electric power plants are rotated because of a flow of steam or water, or wind. Turbines are acting as rotors in the electric power generator, while the electricity is produced in stators based on the principles of electromagnetic induction.

Hydroelectric Stations

Let's estimate theoretically an amount of energy that a hydroelectric station can produce.

First important component in generating electricity at a hydroelectric station is falling water. We can have falling water by building a dam on a river, like Hoover Dam on Colorado river or use natural difference in the level of water of the waterfalls, like at Niagara falls.

Having the water at two levels, we should direct the flow from top to bottom onto turbines through pipes. Amount of electricity that can be generated obviously depends on the amount of potential energy water at the top has relatively to the bottom level. As the water falls through the pipes onto turbine, its potential energy is converted into kinetic energy of moving water. This is how much energy we can use to generate electricity. It depends on the amount of water flowing through pipes and the height difference between the top and the bottom levels.

Let's assume that the amount of water falling down through pipes from top to bottom level is M (kg/sec) and the difference in height from top to bottom is H (m).
That means that the amount of energy falling water is losing per unit of time is
Pwater = M·g·H (J/sec or W)

Then we have to solve a purely technical problem to convert this energy into rotational energy of turbines.

Different designs of turbines have been used and tested during a long time of using hydroelectric power. Contemporary turbines are pretty efficient in this process of conversion, but still far less than 100% effective. Losses of energy always exist, and we need some coefficient of efficiency of a turbine to get exact amount of rotational energy produced by falling water.
Let's assume that k is such a coefficient. It has a value from 0 (absolutely ineffective conversion) to 1 (full energy amount of falling water is converted into rotational energy of a turbine). Then the amount of rotational energy produced by turbines per unit of time is
Pturbine = k·M·g·H

Next step is to convert rotational energy of turbines into electric energy.
This is done by generators, which we discussed in previous lectures.

The contemporary generators are pretty effective with norm being above 90%, so we can assume that the coefficient of effectiveness k introduced above encompasses both effectiveness of converting energy of falling water into rotation of turbines and conversion of rotation into electric energy.
So, overall energy produced by a hydroelectric power station per unit of time (that is, the power produced) is
P = k·M·g·H
The hydroelectric power stations can be very large and can produce a lot of electric energy. The most powerful electric power stations are hydroelectric. The problem is, there are not too many rivers suitable for building hydroelectric power stations and an environmental effect of building a hydroelectric power station can be significant.

At the same time, the hydroelectric power stations are pretty efficient, the coefficient k in the formula above can be above 0.8, which means that about 80% of the power of water falling on turbines is effectively converted into electric power.

Coal Burning Stations

Almost a third of electricity generated in the world is produced by fossil fuel burning power stations.
Let's examine the coal burning power station.

The main steps of producing electricity by burning fossil fuel are
(a) burning fossil fuel to boil water, converting chemical energy of burning fuel into kinetic energy of produced steam,
(b) converting kinetic energy of steam into rotation of turbines,
(c) converting rotational energy of turbines into electricity by generators.

Coal is a major source of fossil fuel with natural gas and oil following.
Convenience of putting a coal burning electric power station anywhere should be weighed against environmental impact of such a plant.

Producing energy from burning coal is not a very efficient way to extract chemical energy. Significant portion of the energy produced by burning coal is wasted on each step and the overall efficiency of such a power station is about 40%. Most of the energy losses occur during the first stage of generating electricity - burning coal to boil water and produce steam.
Some efficiency can be achieved by pulverizing coal to powder. However, the main product of burning fossil fuel - carbon dioxide CO2 - produces some unavoidable negative environmental effect.

Nuclear Power Plants

The difference between a nuclear power plant and coal burning one is at the first stage to boil the water. While at coal burning plants the source of heat to boil water is burning coal, at the nuclear power plant the source of heat is energy released by breaking nuclei of heavy elements, like Uranium or Plutonium, into lighter components using neutrons.

The main steps of producing electricity in a nuclear power plant are
(a) bombarding the enriched radioactive material (Uranium, Plutonium or other) with neutrons causing the nuclei of this material to break, releasing certain amount of heat to boil water getting steam,
(b) converting kinetic energy of steam into rotation of turbines,
(c) converting rotational energy of turbines into electricity by generators.

Efficiency of nuclear power plants is quite limited inasmuch as in coal burning power stations and is about 40%. That is, about 40% of the energy generated by heat is converted into electricity.

Of interest is a process of nuclear fission that produces the heat. Here is a simplified model of this process.

When a nucleus of Uranium-235 (92 protons + 143 neutrons) is bombarded with a neutron, it temporarily accepts this neutron inside, becoming Uranium-236 (92 protons + 144 neutrons).
This isotope is not stable and a nucleus breaks into different parts. This is a complex process and parts might be different.

A typical scenario might be as follows.
Broken parts are Barium (56 protons + 83 neutrons), Krypton (36 protons + 58 neutrons) and 3 neutrons are released to bombard other nuclei of Uranium 235, causing a chain reaction.

The combined mass of all parts is less than the mass of initial components. The remaining mass of an unstable nucleus of Uranium-236 is converted into radiation (heat, gamma-rays). The heat is used to boil the water, converting it into high energy steam to rotate the turbines.

The corresponding equations describing nuclear fission is:
1n0 + 235U92236U92
139Ba56 + 94Kr36 + 31n0 + γ

In reality the process is much more complex because the broken parts of a nucleus might be different, themselves not stable and further emitting elementary particles.
The process must be controlled by reducing the number of neutrons flying in all the directions after fission to prevent a nuclear explosion.