Show that Ampere's Law is not necessarily consistent if the...

Key Concepts

Faraday’s Law of Induction

Faraday’s Law of Induction, which relates a changing magnetic flux through a circuit to the induced electromotive force (EMF), does not suffer from the same inconsistency issue as Ampere’s Law for closed surfaces. Since Faraday’s Law naturally incorporates the phenomenon of electromagnetic induction due to a time-varying magnetic field, it remains consistent regardless of the surface used for evaluating the magnetic flux.

Displacement Current

Displacement current is a term added by Maxwell to account for the contribution of time-varying electric fields to the magnetic field. It plays a crucial role in ensuring that Maxwell-Ampere Law holds true regardless of the shape or closure of the surface used for flux calculations, thus preserving the consistency of the underlying electromagnetic theory.

Ampere’s Law

Ampere’s Law, in its original form, relates the line integral of the magnetic field around a closed loop to the electric current passing through the surface bounded by that loop. However, when applied to a closed surface, ambiguity arises because the original law does not account for the possibility of changing electric fields threading through the surface, leading to inconsistency in how the magnetic field is derived depending on the surface chosen.

Maxwell-Ampere Law

The Maxwell-Ampere Law is an extended version of Ampere’s Law that includes the displacement current term. This modification was introduced by Maxwell to resolve the inconsistency present in the original law when dealing with time-varying electric fields. By adding the displacement current, the law becomes consistent and independent of the particular surface chosen, thereby aligning with the continuity equation and charge conservation.

Transcript

00:01 Now, for this question, i'd like her to think about something as we get into the question.

00:06 Amper's law can be written in terms of total current that includes both free and bound current.

00:13 Also know that it is a line integral of the magnetic field b.

00:17 So it is that line integral of the magnetic field, magnetic field, that magnetic field is represented by b.

00:32 Around the close curve which is proportional to the current passing through the surface s and that can be simply written as this so the ampest law is what we are referring to so we have b d s all equal to mu not from s j d s and when you simplify this they are going to get mu nots from s jds and when you simplify this they are going to get mu nots i am closed i am closed here so here when we talk about this we are saying that the total current the total current density the j here represents the total current density the j here represents the total current density the total current density okay so so take notes of the end, that represents the net charge transport out of the surface.

01:49 Now that we have this in mind, now that we have this in mind, we can start solving this particular question using the formulas that we have generated here.

02:01 So we can get right into solving the question.

02:03 Here, if the charge is inside the region, for example, the charge inside that enclosed region is stationary, then the equation above, this equation, this equation, is going to be.

02:14 To be it's going to be this b dot d s is equal to mu not mule not j the s for the ampers law i enclosed and this is going to be mu not the q all over the t and this is going to give us zero is going to give us zero so take note of that now another thing you should notice that similarly if the charge is in a dynamic condition then you are going to have a different thing so if it's in dynamic condition i don't write everything that i just said that is why i'm just putting the dynamic condition so if the charge is in a dynamic condition then how is it going to be like it's going to be like this it's going to be like this and i'll like you to please take note of that it's going to be like this this is the mule new not sorry new not from s and this is j letter j with the current density j d s and this will be equal to is the same thing like the first one but this time it's not going to be equal to zero it's not going to be equal to zero and also when you look up here it was possible but this time going to be a negative going to be negative all over the t all over the t and this is not going to be equal to zero okay now um when you look at this it just concludes that the ampest floor is not necessarily consistent if the surface through where the flux is to be calculated is a closed surface so this actually answers some part of our question for us the amperse law is not necessarily consistent if the surface through which the flux is to be calculated is a closed surface.

05:05 So we'll move on to the next one.

05:09 We move on to the next one...

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