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Coulomb's Electric Force Law Charles Coulomb set out to determine the precise nature of the electric force law in 1795. Remarkably, despite using primitive tools, he did indeed succeed in formulating this law through experiment. There are two things to consider when dealing with the electric force law just as with the gravitational law that Newton formulated almost a hundred years before Coulomb. Question Six: What is the relationship between electric force and the product of charges? Question Seven: What is the relationship between electric force and the distance between the charges?
These two questions are the exact same we dealt with when we considered gravity. One might be tempted to think
that the relationships between two charges and their force is exactly the same as the
relationship between two masses and gravitational force. If one were to conduct an experiment to evaluate the above
two questions, and then graph the results, you would have the following:
 Interestingly, the electric force does indeed have the exact same relationship that gravity has. Here is the electric force law expressed in words: The electric force is directly proportional to the product of charges, and inversely proportional to the distance between the charges squared.
Expressing the above sentence in equation form yields:
 (the letter q is the symbol used for charge) Again, there needs to be a constanst in order for the proportionality to be expressed as an equation. In the case of electrical force, the constant is given the letter k and equals approximately 9(10^9).  Therefore, Coulomb's electric force equation is written as,  Question Eight: Of the four fundamental forces in the Universe, which one is the stronges? The weakest? There are four fundamental forces in the Universe. They are as follows:
1. Strong Nuclear Force:
This force is responsible for keeping the nucleus of the atom from
breaking apart. If you have not already realized, the nucleus consists of like charges, the protons, being
crammed together into a very tiny space. In fact, the density of the nucleus is 1(10^14) g/cm^3 (read grams per cubic centimeter). To
appreciate the enormity of this density, if a penny were to be formed with the same exact
density, it would weigh in excess of 30 million tons! So, because there are like charges
being compacted together, there must be something to keep them from flying off from one
another. That something is the Strong Nuclear Force.
2. Weak Nuclear Force:
This force governs the radioactive decay of atoms.
3. Gravitational Force:
This force is responsible for keeping you planted in your chair, keeping
the planets in orbit around the sun, and creating fascinating objects like black holes and
neutron stars.
4. Electro-Magnetic Force:
Technically speaking, the electric force is only half the story. It
turns out that with electrical comes magnetic. (More on that later). This chapter is only
dealing with the electrical part of this force and is responsible for the force between
charged objects.
If we consider a proton and an electron .005 meters apart, we can find the gravitational force
between them by using the Universal Gravitational Equation:
 Which equals 4.07(10^-63) Newtons! The electric force can be found similarily,  Which equals 9.22(10^-24) Newtons. Obviously both forces are extremely tiny, and from the answers, it is clear that the electric force is greater than the gravitation force. However, the degree to which how much bigger is staggering. If we take the electric force and divide by the gravitational force to find how much bigger the electric force is, we get an almost unbelievable number!  That is 226 followed by 37 zeros! To truly appreciate the enormous nature of this number, I always write it out with all of the zeros in class. The actual name for this number is 2.26 duodecillion! Brian Greene, author of The Elegant Universe gives a remarkable analogy. If your right bicep represents the strength of gravity, your left bicep would have to be the size of the known Universe to represent the strength of the electrical force! You can no doubt see that the graviational constant is much smaller than the electrical force constant. In fact, the electrical force constant is 1.35(10^20) times bigger! But how is it that gravity is that much weaker when it is able to hold such massive planets in their place? That is the peculiar thing here. Look at it in this perspective. It takes an enormous amount of mass just to create enough gravity to keep us firmly planted on Earth. The Earth's mass is 5.98(10^24) kilograms. That is 6,608,000,000,000,000,000,000 tons! If the gravitational force was as strong as the electrical, we would weigh 2.26 duodecillion times our current weight! This does bring up another question. Question Nine: If the electrical force is that much stronger than gravity, why doesn't it dominate our everyday life?
This is a fair question to ask, and it has a simple answer. Most things in the Universe are
electrically neutral. This is due to the fact there is both a positive and negative charge,
so they can cancel out. Gravity never cancels and is always attractive, so it dominates our
everyday life. Remember earlier we said most objects only obtain a tiny fraction of
a coulumb of charge. Again, if you had two objects with one coulomb of charge each
attracting one another from a distance of 1.0 meter, the force between them would be,
Which equals approximately 1,012,000 tons! So, it is good that most objects are electrically neutral, and when they are charged, they are still close to neutral because of the tiny amount of net Coulombs. OK, we need to answer question eight. Here is the order of the four forces from strongest to weakest:
1. Strong Nuclear
2. Electro-Magnetic
3. Weak Nuclear
4. Gravity
It is like the tiny weight lifter compared to the large weightlifter. The large weightlifter might be able to lift more weight than the tiny one, just like Gravity deals with larger objects. Let's say he is able to bench press 400 pounds, and weighs 300 pounds. That means he is able to lift 1 and 1/3 times his weight. On the other hand , the tiny weightlifter may be able to lift only 300 pounds, but he weighs 150 pounds. Therefore, he can actually lift two times his weight. Pound for pound, the tiny person is stronger than the larger one even though he is unable to lift what the larger one can.
Here is the fascinating thing about all four forces. If their strengths were any different,
such as depending on the inverse of the distance and not the inverse square of the distance, or the constants were different,
the Universe as we know it would not exist. Atoms wouldn't be able to stay together so Stars
wouldn't be able to form, which means there couldn't be any planets, and ultimately no existence
of life. The Universe is the way it is because of the precise nature of the four forces of
nature. It turns out that our Universe is extremely unique and precisely the way it is so
that it can exist the way it does. There are so many parameters to its makeup that if they
were even slightly different, we would not be here!
For hundreds of years, this question went unanswered. The prevailing idea was called Action at a Distance. This means that the gravitational force simply exerted itself over a distance instantaneously, and didn't need contact between objects to work. Not much of an explanation. Nonetheless, it was the accepted idea. In fact, it was the idea that explained the electrical force as well. It took the insight of a man that wasn't actually all that gifted mathematically, but he had imagination. In 1831, Michael Faraday proposed what is now called Field Theory. It simply says that a charge creates an electric field that surrounds it and emanates in all directions. When another charge comes into that field, it is the field that exerts the electric force. So, his field theory only addressed the electric force, but it turns out that it was the correct assumption for gravity as well. However, it would take another 84 years before someone described that field with mathematical precision. That person was Albert Einstein. Here is a brief explanation of the gravitation field.
In order to explain this idea, we need to consider space as a two dimensional flat piece of
stretched rubber, like a trampoline. When there isn't any mass, space is flat.
 When a mass is placed in that space, such as the Earth, the space is affected, much like a trampoline would be indented if you placed a bowling ball at its center.  Now when another mass is within the vicinity of the Earth, the field exerts a force on that mass, and keeps it on the ground, or in orbit. If you look at the picture above, you get the sense that the ball would slide down the indention towards the Earth. That actually isn't too far from the truth, although highly simplified. This is the essence of the General Theory of Relativity, which Einstein published in 1915. I will go in more depth with this topic in the Relativity section of the website after I finish this book. Back to Faraday's field for electric charges. Like I mentioned before, Faraday was not a gifted mathematician. In fact, his field theory idea didn't have a solid mathematical foundation. However, in 1865, James Clerk Maxwell developed a comprehensive mathematical theory that explained everything there was to the electro-magnetic realm. Within this theory contained Coulomb's electric force law, and a true mathematical description of Faraday's field theory. So, it wasn't just a good idea, it was the right idea. This just goes to show you that it doesn't always take a mathematical genius to make a powerfull contribution to the advancement of knowledge for mankind. Now the question becomes: what does the mathematical model say about the elecric field? |