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#### The Speed of Electricity

By Archis Gore, Masters in Scientific Computing, University of Pune.

I have always noticed how many of our students are ignorant on the very basis of what science is. There are people who know a lot about a lot but are totally ignorant on very basic issues. The educational system is partly a reason for this and the lack of students curiosity is the other.

Currently trends are towards increasing creativity amongst students. But what we lack is curiosity. Before creativity, there must be curiosity. There must be a motivation to know something just for the sake of it.

The concept of the speed of electricity is a very common example that I use to confuse many of my peers. And many people are surprised of how totally distant their guesses are from my estimates. During this article, we shall try to calculate the speed of electricity ourselves to get a general idea of the order of magnitude.

I am going to ask you to do something that I always ask all the people to whom I explain this concept. Please try to make an estimate of the speed of electricity after you have read the specifications of the system described below. This estimate must be made orally and just an order of magnitude would be fine. Many answers that I generally get to this question is the speed of light. Some even go on to comment that since they have learned that the speed of light is the absolute limit to how fast anything can travel, they are stopping short of naming a higher value.

I must comment that the speed of electricity is a very subjective issue based on the current, conduction and resistance of the circuit. So I am considering the following well defined system.

Let us consider a common light bulb working on DC (direct current, this will simplify calculations). Let us assume the household voltage of 250 volts, and a standard copper wire of radius 1mm connecting the bulb serially to the voltage source. Assume also, that the bulb consumes 40 watts of power. This rating is decided by the manufacturing company considering the resistance of the bulb and the voltage that is going to be applied to it. This eliminates the need for us to know the resistance of the bulb. Let us assume for the sake of convenience that the copper wire has zero resistance (this will in fact give us a higher estimate for the speed of our electrons).

Now let us first clearly define the speed of electricity that we mean. We shall try to estimate the average distance covered by an electron in the wire in one second when the entire circuit is closed.

First lets calculate the total current that will flow through the bulb. Since we have a voltage of 250 volts and a wattage of 40 watts, the current should be equal to power/voltage=40/250=0.16A.

Now lets divert to another angle. Let us consider a 1cm long segment of the wire used in our circuit. We shall try to estimate the total charge that that segment of wire holds at any instant. We can find the volume of the segment as follows:

Volume=P * r2 * l

Where r is the radius of the wire and l is the length of the segment.

Hence, the volume comes out to be (in cubic metres):

Volume=3.14 * (0.001)2 * (0.01) = 3.14 * 10-6 * 10-2 = 3.14 * 10-8 cubic metres

Now we consider the average number of atoms of copper that this segment of wire must be holding. The density of copper is 8.96 kilograms per cubic metre and its atomic weight is 63.546. Atomic number of copper is 29 which we will require later on in our calculations.

Hence our segment of wire weighs approximately:

Mass=3.14 * 10-8 * 8.96 kilograms = 2.81344 * 10-7 kilograms = 2.81344 * 10-4 grams

Generally 1 atomic mass unit = 1.6603 * 10-24 grams. Since the atomic weight of copper is 63.546, the weight of one atom of copper in grams is given by:

Weight of 1 atom of copper = (1.6603 * 10-24 * 63.546 ) grams = 105.5054238 * 10-24 g

= 1.055054238 * 10-25 kg

Now, we must find out how many atoms of copper are crammer up into 2.81344 * 10-7 kg of copper. We simply divide the total weight of the copper segment by the weight of an individual atom of copper as follows:

Number of copper atoms in segment=2.81344*10-7/1.055054238*10-25 = 2.66663 * 1018

Just to be sure, we shall calculate the same using a different approach. Avogadros number gives us the number of molecules (in case of copper, the number of atoms) of a specific substance contained in one gram-mole (molecular weight in grams) of that substance. Hence, one gram mole of copper should contain 6.0221367 * 1023 atoms. The molecular weight of copper being 63.546, one gram-mole of copper is 63.546 grams. Now using some middle school algebra, knowing that 63.546 grams of copper contains 6.0221367 * 1023 atoms, we can easily calculate the number of atoms in 2.81344 * 10-7 kilograms (equivalent to 2.81344 * 10-4 grams) of copper. The calculations are shown below:

63.546 grams : 6.0221367 * 1023 atoms

2.81344 * 10-4 grams : ? atoms

x=6.0221367 * 1023 * 2.81344 * 10-4 / 63.546 = 0.2666245 * 1019 atoms

= 2.666245 * 1018 atoms

This matches our estimate and hence we are probably on the right track. We shall now find the average between the two values and consider that average as the number of atoms of copper contained in our segment of wire of radius 1mm and length 1cm. Hence,

Average number of copper atoms in segment are = 2.6664375 * 1018

One atom of copper contains 29 electrons since its atomic number is twenty-nine. Now we know that not all electrons in an atom are free to move while conducting current. This is the part on which I am unsure and am making a safe assumption. The electron configuration of copper is 2-8-18-1. I will assume that only one electron per atom is allowed to move since only one electron is present in the valence band (outer most band). This is actually a more complicated matter involving primary and secondary quantum numbers and sub-orbitals of equal energy and so on but we shall do away with all this. If in fact more electrons are allowed to move in an atom, then our calculations will give a higher value than the actual value. So our value will be a very safe estimate.

Now since the charge on each electron is 1.6021892 * 10-19 coulomb, an atom of copper will carry movable charge equivalent to:

Movable charge on copper atom=1.6021892 * 10-19 coulomb

The charge on the segment of wire under consideration is:

Charge on segment = number of atoms in segment * charge on each atom

= 2.6664375 * 1018 * 1.6021892 * 10-19 C

= 4.272137364975 * 10-1 C

Please note that due to our consideration of only one movable electron, this value of available charge is lower than the actual value. However, please remember that whatever value we calculate for our speed of electricity will be reduced drastically if the actual number of movable electrons turn out to be higher than my assumption.

Now since the current through the bulb is 0.16 A, it means that 0.16 C of charge is traveling through any cross-section of wire at one second. Now we shall calculate how long a wire we require so that it will hold 0.16 coulombs of charge.

This is done as follows:

This is a simple technique learnt in grade school to find out how much 5 mangoes will cost if we know that 7 mangoes cost 35 rupees.

1 cm of wire : 4.272137364975 * 10-1 C

x cm of wire : 0.16 C

Therefore, x= 0.16 / (4.271828943554 * 10-1) = 0.037455 * 10 cm

= 0.37455 cm

For this discussion consider a segment of wire of length = 0.37455 cm. Now since this length of wire holds 0.16 C of charge, it means that when the wire is conducting current and all the electrons at any given instant in the wire go out of one end and new ones enter through the other, we can say that the wire has conducted 0.16 coulombs of charge. Now if this conduction takes place in a second, we can say that the wire has conducted current at the rate of 0.16 A. This means that when the electron on the atom situated at the extreme one end of this segment must completely move to the other end in one second in order for the wire to carry a current of 0.16 A.

Since the wire in our hypothetical scenario is carrying 0.16 A of current, it means that every electron in the wire must move 0.37455 cm each second in order to satisfy the scenario. Let us consider the speed of electricity in the defined system to be approximately 0.4 cm/sec allowing a large margin of error in our calculations which might have made it slower than usual. This means that it would take slightly more than four minutes for each electron to even move one meter. Please note that we have considered a very thin wire in this example and have also made liberal allowances in our calculations. In reality, the speed of electrons is even slower.

This surprises even the geniuses of our schools. Now we can retrospectively analyze why many people think of electricity as having an almost infinite speed even though it is really slower than an old snail.

The process of conducting any scientific experiment involves the following essential steps  observation, conclusion and result. Now in case of the experiment mentioned above, both the steps of the observation and result are perfectly logical, however, it is the conclusion drawn from the observation which makes the result go wrong. When we switch on a light bulb, we instantly see light on our floor. The conclusion most people unconsciously arrive at is that the electricity has traveled from the switch to the bulb in the time it has taken the bulb to light up. In a manner of speaking, this is true since the speed of electricity can be thought of as the speed with which an action performed at one place can produce the intended effect at another place. And that is why I explicitly defined the speed of electricity as the displacement per unit time of an electron in the conducting material at the beginning of this article.

Well, if electricity doesnt travel from the switch to the bulb in such a short time, then how does the bulb light up so quickly? The reason is that electricity is already in the bulb and doesnt need to travel from the switch to the bulb. Electricity or rather electrons are present throughout the conductor. What we really do when we turn on the switch is push the electrons near the switch very slowly. These electrons in turn push their forerunners and so on until the electrons present in the bulb start moving. This is what gives us the illusion that electricity has traveled very fast from the switch to the light bulb.

A good analogy to explain this phenomenon is to consider an apartment building having a height of 30 feet. There is a water tank on top of the building which is filled. Whenever anyone on the ground floor opens their tap, water instantly rushes out of the tap. Does this mean that the water travels more than 30 feet per second through the pipe? It would be a motorists dream to travel at such speeds!

As an end note, I would like to comment that there are so many other such simple secrets in our life which if unlocked will improve our understanding and thinking to greater levels.