Two questions about introductory EM confusions.

In summary, the conversation discusses the definition of electromotive force and potential difference in electrostatic and electrodynamics cases. The question asks if there is a similar reasoning to find the same result for electrodynamics cases where the magnetic force is involved. The answerer mentions the Lorentz force and advises against using the concept of "positive integral of E" due to its complexity. Instead, they suggest studying the Lorentz force as it is fundamental in understanding the relationship between electromotive force and potential difference.
  • #1
Coffee_
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QUESTION 1. I have asked this question here a while before but it seems that I have still not gotten it. My apologies to the previous answerers of this one, I guess it's just me being slow.

We defined the electromotive force ##\epsilon## for a charge that moves along some path while some force ##\vec{F}## of electromagnetic origin acts upon it C as: ##\int_C \frac{\vec{F}}{q}\vec{dl}##

The electrostatic definition of a potential difference between point A and point B for any path C going from A to B is given by ##\Delta{V}=-\int_C \vec{E} \vec{dl}##

It is easy to see that in purely the electrostatic case for a path C from point A to point B ##\Delta{V}=-\epsilon## Is there a way to do an exactly the same formal reasoning to find this exact same result for electrodynamics cases where the magnetic force is involved? Because right now I have the feeling that in electrodynamics the result is going to be ##\Delta{V}=\epsilon## without the minus sign.

QUESTION 2. This question is about a step in a derivation I'm confused about. The point of the derivation is to show that the energy density of a created magnetic field goes with ##B^{2}##. Consider a single loop with a current ##I## running through it. The potential energy stored in the total flux through the surface is equal to ##U=\frac{I \Phi}{2}##

And so we can associate a small potential energy with a small piece of flux through the loop as ##dU=\frac{I d\Phi}{2}## Now let's take the special geometric set of field lines that contain the same amount of field lines through space. I think they are called stream surfaces? Anyway the point is that with each ##d\Phi## through the surface we can associate such a stream surface through which the flux is constant.

This gives ##dU=\frac{I \vec{B} \vec{dS}}{2}##. Where ##\vec{B}## and ##\vec{dS}## can be taken anywhere on the stream surface as long as they are both corresponding to each other. This is because the amount of field lines and thus the flux through a stream surface is constant.

Now we use Maxwell's magnetic circulation law in integral form along one field line in this stream surface. This means that ##\oint_C B dl = \mu_{0} I## where I already use that B and dl are parallel.

So including this into our expression we find: ##dU= \frac{1}{2\mu_{0}} \oint_C B^{2} dl dS(x,y,z)## What basically happens here is that one B field is being integrated over. The other expression has the B field for the flux which can be any B value along the stream surface as long as it corresponds to a correct ##dS## of the stream surface. If I take the value for the latter B field equal to the value of the B field I'm taking a line integral over dS would become a function along the line integral. I get everything until here.

The next step is the proof saying that if we integrate over all those ##dU## we find ##dU= \frac{1}{2\mu_{0}} \iiint_V B^{2} dV## and thus the value in the integral represents the energy density.

I don't get the math or formal reasoning behind the last step, can someone explain or elaborate.

Thanks a lot.

 
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  • #2
For Q1 check out this thread from last Thursday. Pretty hefty stuff. Sorry, no time at the moment for Q2...
 
  • #3
BvU said:
For Q1 check out this thread from last Thursday. Pretty hefty stuff. Sorry, no time at the moment for Q2...

I meant to imply something else with my first question than what is being discussed in that thread. I already broke my head a lot on what they talked about but luckily I already get that more or less. My question is about the fact that to me it seems that the relationship between ##\Delta{V}=-\epsilon## seems to have a different sign in electrostatics and electrodynamics cases. I can show ''formally'' what the sign has to be for electrostatic cases which I did but I wanted to see a similar reasoning for when the magnetic field is involved but seem to not succeed in that.
 
  • #4
Then probably this thread is also beside the point.

Could you elaborate be a bit more on this expectation that ##\Delta{V}=+\epsilon## in electrodynamics cases ? V still the electrostatic potential ? What cases ? Examples ?
 
  • #5
BvU said:
Then probably this thread is also beside the point.

Could you elaborate be a bit more on this expectation that ##\Delta{V}=+\epsilon## in electrodynamics cases ? V still the electrostatic potential ? What cases ? Examples ?

I remember the following from class. This is not a correct reasoning below but this is just an example of which I remember a result along the lines of a positive relationship. Again, it's something from class a while ago and I don't remember how the reasoning went exactly:

Take an open circuit and move it through a B field. The charge accumulation will happen in such a way that the E-field at each point counteracts ''v x B'', so that ##\vec{E}=-\vec{v}\times\vec{B}##. So taking the integral of the E field would be taking MINUS the integral of the resulting magnetic force per charge. So this has the form of ''positive integral of E, gives MINUS the work per charge by the magnetic force''.

A positive integral of E also gives MINUS the potential difference. And from such a reasoning my suspicion arsises. I can't really put a finger on it hence I'm asking if someone can provide a thorough reasoning along these lines.
 
  • #6
The thread in post #4 IS beside the point -- either that or way too complicated. You're much better off studying the Lorentz force. It's very fundamental.

(The following is not very authorative, just my way of looking at things) :

I see how you work ##\vec v \times \vec B## to something with an ##\vec E## character, and then the countertaction into a minus sign. But I would advise against it:
  • ##F_{Lorentz} \perp \vec B## so the B field itself does NOT do any work. The work is done by whatever brings about ##\vec v##.
  • ##F_{Lorentz}## is the cause of charge accumulation, which in turn causes an E-field, which you can work around to a potential outside the conductor. Inside the conductor only if it has non-zero resistance! So if you close the loop outside the B field you can get a current. (Also if you rotate a loop within the B field, because then you change the direction of ##\vec v##). From these Lorentz force consequences the Faraday law can be derived. (I like the counterexamples :) -- there the Lorentz force pops up again, and "wins" -- but not to worry: the Maxwell-Faraday equation is always correct)
 
  • #7
BvU said:
The thread in post #4 IS beside the point -- either that or way too complicated. You're much better off studying the Lorentz force. It's very fundamental.

(The following is not very authorative, just my way of looking at things) :

I see how you work ##\vec v \times \vec B## to something with an ##\vec E## character, and then the countertaction into a minus sign. But I would advise against it:
  • ##F_{Lorentz} \perp \vec B## so the B field itself does NOT do any work. The work is done by whatever brings about ##\vec v##.
  • ##F_{Lorentz}## is the cause of charge accumulation, which in turn causes an E-field, which you can work around to a potential outside the conductor. Inside the conductor only if it has non-zero resistance! So if you close the loop outside the B field you can get a current. (Also if you rotate a loop within the B field, because then you change the direction of ##\vec v##). From these Lorentz force consequences the Faraday law can be derived. (I like the counterexamples :) -- there the Lorentz force pops up again, and "wins" -- but not to worry: the Maxwell-Faraday equation is always correct)

I see, thanks for your time and help!
 

1. What is introductory EM?

Introductory EM (electromagnetism) is a branch of physics that deals with the study of electromagnetic fields and their effects on matter. It is typically studied in the first or second year of undergraduate physics courses and is essential for understanding a wide range of phenomena, from electricity and magnetism to light and radio waves.

2. What are some common misconceptions about introductory EM?

Some common misconceptions about introductory EM include the belief that electric and magnetic fields are separate entities, when in fact they are interrelated and form the foundation of electromagnetism. Another misconception is that electric fields only exist in the presence of a static charge, when in reality they can also be created by changing magnetic fields. Students also often struggle with the concept of electromagnetic induction and how it relates to generators and transformers.

3. How can I better understand the concept of electric and magnetic fields?

A helpful way to understand electric and magnetic fields is to think of them as invisible forces that surround charged particles and magnets, respectively. These fields influence the behavior of other charged particles and magnets within their range, and can be visualized using field lines. Additionally, practicing problems and working through examples can also aid in understanding these concepts.

4. What are some real-life applications of introductory EM?

Introductory EM has numerous real-life applications, including the functioning of electronic devices such as computers, smartphones, and televisions. It also plays a crucial role in the development of technologies like MRI machines, which use magnetic fields to create images of internal body structures. Furthermore, understanding introductory EM is essential for the study of electricity and power generation, telecommunications, and many other fields.

5. What are some tips for succeeding in an introductory EM course?

Some tips for succeeding in an introductory EM course include staying organized and keeping up with readings and assignments, seeking help from professors or tutors when needed, and actively engaging in class discussions and problem-solving. It can also be helpful to form study groups with classmates and to practice problems regularly to solidify understanding. Additionally, approaching the subject with curiosity and an open mind can make the learning process more enjoyable and effective.

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