Reason for Opposite Signs for Terms in Faraday's Law and Ampere's Law

[jbrandew]

TL;DR

Confused about intertial response

Hello, im a bit confused about the right term in Faradays Law vs the right most term in Amperes Law. They both seem like terms corresponding to a resistance to change, similar to inertia. As in, the induced electric field from a changing magnetic field has an opposite direction compared to the change. It seems like a similar concept in the case of Amperes Law, where we have the displacement current term.

However, the displacement current term has the same sign as the left side magnetic field. Why is this? Shouldn't it also be negative to correspond to the opposite direction of the magnetic field?

Thank you!

 

[anuttarasammyak]

See Figure 18-2 in https://www.feynmanlectures.caltech.edu/II_18.html .
It would convince you plus sign of dispace current term in Maxwell equation.

 

[Sagittarius A-Star]

TL;DR Summary: Confused about intertial response

Hello, im a bit confused about the right term in Faradays Law vs the right most term in Amperes Law. They both seem like terms corresponding to a resistance to change, similar to inertia. As in, the induced electric field from a changing magnetic field has an opposite direction compared to the change. It seems like a similar concept in the case of Amperes Law, where we have the displacement current term.

However, the displacement current term has the same sign as the left side magnetic field. Why is this? Shouldn't it also be negative to correspond to the opposite direction of the magnetic field?

Thank you!

A formal interchange $B \mapsto E$ and $E \mapsto -B$ (in Gaussian units) leaves the relativistic transformation of the electromagnetic field components invariant.

The same formal interchange converts i.e. Amperes Law into Faradays Law, if also the electric current density is replaced by minus the magnetic monopole-current density
$j_e \mapsto -j_m$, which is assumed to be zero.

 

[vanhees71]

That's called the duality symmetry of electrodynamics. It's of course only a complete symmetry, if there are magnetic monopoles. Unfortunately there's no evidence for (elementary) magnetic monopoles. They have been observed only in the sense of quasiparticles in condensed matter physics of "exotic" matter called "spin ice".

 

[Gavran]

In the equation $$ \vec \nabla \times \vec E = - \frac { \partial \vec B } { \partial t } $$ an electric field can force free electric charges to move, producing an electric current which will generate a magnetic field. This generated magnetic field will oppose the changes in the initial magnetic field and there is a minus sign in the equation.

In the equation $$ \vec \nabla \times \vec B = - \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ a magnetic field can force free magnetic monopoles to move, producing a magnetic current which will generate an electric field. This generated electric field will oppose the changes in the initial electric field and there is a minus sign in the equation. Of course this is not true. Magnetic monopoles do not exist and a magnetic current does not exist too. So there will be nothing to generate an electric field which will oppose the changes in the initial electric field and there is a plus sign in the equation. The equation $$ \vec \nabla \times \vec B = \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ is a correct equation.

 

[renormalize]

This generated electric field will oppose the changes in the initial electric field and there is a minus sign in the equation. Of course this is not true. Magnetic monopoles do not exist and a magnetic current does not exist too. So there will be nothing to generate an electric field which will oppose the changes in the initial electric field and there is a plus sign in the equation.

Are you claiming here that the arrangement of the signs in Maxwell's equations somehow rules out the existence of magnetic monopoles? It's perfectly possible to generalize Maxwell's theory to include magnetic charge- and current-densities (https://en.wikipedia.org/wiki/Magnetic_monopole):

Based on our current understanding, the apparent absence of magnetic monopoles in nature is an experimental observation, not a theoretical mandate.

 

[Sagittarius A-Star]

In the equation $$ \vec \nabla \times \vec B = - \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ a magnetic field can force free magnetic monopoles to move, producing a magnetic current which will generate an electric field. This generated electric field will oppose the changes in the initial electric field and there is a minus sign in the equation.

No. Here you would need a plus sign, because the curl E in the extended Faraday's law has already the opposite sign as the magnetic current, see screenshot in posting #6 of @renormalize.

 

[Gavran]

In regions where $$ \vec \nabla \cdot \vec B = 0 $$ holds, magnetic monopoles do not exist. There are not regions where $$ \vec \nabla \cdot \vec B \neq 0 $$ holds and it can be said a zero in the equation $$ \vec \nabla \cdot \vec B = 0 $$ rules out the existence of magnetic monopoles.

The same thing can be said for the equation $$ \vec \nabla \times \vec B = + \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ where in the case of regions with magnetic monopoles a plus sign in the equation would break the law of conservation of energy. There are not regions where the equation $$ \vec \nabla \times \vec B = + \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ does not hold and it can be said a plus sign in the equation rules out the existence of magnetic monopoles.

 

[renormalize]

My apologies, but I still don't follow your argument. Can you please clarify: in your opinion, can the generalized Maxwell equations shown in post #6 (which include magnetic charge- and current densities) be valid and consistent in a hypothetical universe that does contain magnetic monopoles, or do you believe them to be somehow inherently inconsistent, thereby ruling-out such a universe?

 

[Hornbein]

I used geometric algebra for such purposes. In that way of thinking there is no such thing as a magnetic pole at all, so a monopole is out of the question.

The pole concept is a byproduct of the cross product, an artificial mathematical convenience. The geometric algebra method is mathematically equivalent though slightly more verbose. I used it because I was working in an imaginary 4D universe so there wasn't much choice. The cross product works only in 3D while geometric algebra is applicable in any number of Euclidian dimensions. There the magnetic force at any point is a plane, a magnitude, and a sign, the result being a curved/circular motion in that plane. The sign denotes the direction of the curve.

Well, in 4D there COULD be a 2D pole and a cross product but that would be more verbose and inconvenient and unnatural than geometric algebra so no one would do it that way.

 

[Sagittarius A-Star]

The same thing can be said for the equation $$ \vec \nabla \times \vec B = + \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } $$ where in the case of regions with magnetic monopoles a plus sign in the equation would break the law of conservation of energy.

No, it would not break the law of conservation of energy because the minus-sign is already included in the generation of closed electric field lines around a magnetic monopole-current. For this you would apply the left-hand grip rule:

Source:
https://en.wikipedia.org/wiki/Magne...d_due_to_moving_charges_and_electric_currents

See also the signs in the extended Faraday's law in posting #6.

 

[renormalize]

I used geometric algebra for such purposes. In that way of thinking there is no such thing as a magnetic pole at all, so a monopole is out of the question.

I'd like to read more about this. Can you suggest a published reference that explicitly demonstrates that geometric algebra excludes the existence of magnetic monopoles?

 

[Hornbein]

I'd like to read more about this. Can you suggest a published reference that explicitly demonstrates that geometric algebra excludes the existence of magnetic monopoles?

No, I've never seen the issue discussed. But it seems to me that geometric algebra is just a system of notation and calculation. It can't exclude magnetic monopoles. If they exist they exist.

 

[renormalize]

No, I've never seen the issue discussed. I have a book (8,000 miles away) that has Maxwell's equations in geometric algebra, which have a sign flipped. They didn't discuss this esoteric topic.

But in post #10 you declared:

I used geometric algebra for such purposes. In that way of thinking there is no such thing as a magnetic pole at all, so a monopole is out of the question.

Isn't this statement just personal speculation if you've never seen it discussed anywhere else?

 

[Hornbein]

But in post #10 you declared:

Isn't this statement just personal speculation if you've never seen it discussed anywhere else?

I've said all I'm going to say about this.

 

[Gavran]

It is clear now.

The magnetic current is a fictitious current, but if it exists the magnetic current density in the equation $$ \vec \nabla \times \vec E = - \frac { \partial \vec B } { \partial t } - \mu _ 0 \vec { J _ m } $$ must be preceded by a minus sign because in the equation $$ \vec \nabla \times \vec B = \mu _ 0 \varepsilon _ 0 \frac { \partial \vec E } { \partial t } + \mu _ 0 \vec { J _ e } $$ the electric field generated by the magnetic current must oppose the changes in the initial electric field.

There is the right hand grip rule for the direction of the magnetic field generated by an electric current and there is the left hand grip rule for the direction of the electric field generated by a magnetic current. These two rules must be opposed to each other because they can not be in confrontation with the law of conservation of energy.