Understanding Work Done and Integration: Exploring Small Work and Electric Flux

In summary, the reason why small work done, dW, is taken as F.dx instead of x.dF is because integrating the latter does not give the same result. This is because x often cannot be written as a function of F. Additionally, displacement is not always a function of force, so x.dF is not always meaningful. The area made with the displacement axis is the only relevant area for analyzing physics.
  • #106
Mr Real said:
Yes, I have got several good answers and they have partially answered my questions, but not completely.
I don't see any question of yours which has been partially answered.

1) You wanted to know why the integral of x.dF did not also give work just like F.dx. The complete answer was that it's a calculus error to switch the quantities like that. The area under a curve is not the same as the area to the left of the curve. So whatever x.dF might represent it is not generally work.

2. You wanted to know why textbooks don't spend time discussing this. The complete answer is that there are an infinite number of mistakes you could make, and this is a very rare one. Authors focus primarily on the correct ways, not one of the infinite incorrect ways. When they do address mistakes, it is only the common ones.

3. You wanted to know why the integral of x.dF is not useful in its own right. The complete answer is twofold, first, it is just an odd expression. It is not common to express x as a function of F, and it is often not even possible to do so, and the quantity dF is a weird quantity without physical use elsewhere. Second, there is no known relationship of that quantity to other known physical quantities of interest, nor does it posses known properties that make it a primary quantity of interest.

What could possibly be considered incomplete about this discussion? Not only has this community correctly and completely answered your question, we have been exceptionally patient with you. This discussion should have ended on the first page as soon as it was pointed out that it is a mathematical mistake to switch the terms that way. So your answer has been correct, complete, and exceptionally supportive.

It is highly frustrating to have such thoughtful and helpful replies from the community regarded as insufficient.

At this point I am going to close this thread. I only hope that in the future you can recognize when you have received such high quality responses, and be a little more encouraging to the community that provided them.
 
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<h2>1. What is work done in physics?</h2><p>Work done in physics is defined as the product of the force applied on an object and the displacement of the object in the direction of the force. It is a measure of the energy transferred to or from an object by means of a force acting on the object.</p><h2>2. How is work done related to integration?</h2><p>Work done is related to integration through the concept of a path integral, which is a mathematical tool used to calculate the total work done by a force over a given path. Integration is used to sum up all the infinitesimal work done along the path, resulting in the total work done.</p><h2>3. What is small work in physics?</h2><p>Small work in physics refers to the work done by a force over a very small displacement. It is often used in the context of calculating the work done by a varying force, where the force changes over a small distance and can be approximated as constant.</p><h2>4. How is electric flux related to work done?</h2><p>Electric flux is related to work done through the concept of electric potential energy. The work done by an electric field on a charged particle is equal to the change in its electric potential energy. Electric flux is a measure of the number of electric field lines passing through a given area, and it is related to the electric potential energy through integration.</p><h2>5. Why is understanding work done and integration important in physics?</h2><p>Understanding work done and integration is important in physics because it allows us to calculate the total amount of energy transferred to or from an object by a force. This is essential in many areas of physics, such as mechanics, electromagnetism, and thermodynamics, where the concept of work is fundamental in understanding the behavior of physical systems.</p>

1. What is work done in physics?

Work done in physics is defined as the product of the force applied on an object and the displacement of the object in the direction of the force. It is a measure of the energy transferred to or from an object by means of a force acting on the object.

2. How is work done related to integration?

Work done is related to integration through the concept of a path integral, which is a mathematical tool used to calculate the total work done by a force over a given path. Integration is used to sum up all the infinitesimal work done along the path, resulting in the total work done.

3. What is small work in physics?

Small work in physics refers to the work done by a force over a very small displacement. It is often used in the context of calculating the work done by a varying force, where the force changes over a small distance and can be approximated as constant.

4. How is electric flux related to work done?

Electric flux is related to work done through the concept of electric potential energy. The work done by an electric field on a charged particle is equal to the change in its electric potential energy. Electric flux is a measure of the number of electric field lines passing through a given area, and it is related to the electric potential energy through integration.

5. Why is understanding work done and integration important in physics?

Understanding work done and integration is important in physics because it allows us to calculate the total amount of energy transferred to or from an object by a force. This is essential in many areas of physics, such as mechanics, electromagnetism, and thermodynamics, where the concept of work is fundamental in understanding the behavior of physical systems.

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