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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}##

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 eachother. 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

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

Thanks a lot.

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 eachother. 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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