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Before answering the question, let's talk about the concepts of fields, conservative forces, and conservative fields.

Field refers to the distribution of a physical quantity in space, which tells us that at a certain time, the physical quantity is only related to spatial coordinates and is not affected by other factors.

This physical quantity can be scalar or vector. For example, the distribution of temperature is a scalar field. The spatial distribution of some forces can form vector fields.

For example, the gravitational acceleration of an object of unit mass at different places is a spatial distribution that forms a field called gravity field.

Electric and magnetic fields, for example, are typical fields.

The action caused by the field, that is, the force, can itself form a field, called a force field. The condition is simple, as long as the force at any point in space can be expressed as such a vector function at a certain time.

For example, the spatial distribution of the electric field forces on particles with a definite charge forms a force field.

But the magnetic forces on charged particles are not fields, because they depend not only on spatial coordinates but also on the velocity (magnitude and direction) of the particles, which is not necessarily a function of space.

Moreover, it is easy to imagine that forces such as friction, centripetal force, and Coriolis force cannot form force fields because their magnitude and direction are not determined by spatial coordinates.

So, what is a conservative field?

I believe most people have not seen the definition of conservative field, but most people should know what conservative force is, so let's talk about conservative force first.

What is conservative force? A conservative force is a force whose work is path independent, i.e., whose integral depends only on the coordinates of a and b and is independent of the specific path of integration.

Obviously, if the force itself is not a force field, it must also contain variables outside space, which must be preserved after spatial integration, and the result will not depend only on the starting and ending positions in any case.

Thus, to be conservative, the force must itself be a force field.

So, if you ask why Lorentz force is not conservative? The answer is: because it's not even a force field! Otherwise, you'd be confused. After all, Lorentz force really does work independently of the path-it doesn't do work at all!

How practical is this definition of conservative force?

By definition, this is an unprovable proposition, because you cannot test all possible paths. Of course, if you find that different paths between two points lead to different work, then it must not be conservative.

Mathematics, however, provides us with a test: a force is conservative if and only if there is a gradient of a function such that it can be expressed as.

The reason why we say "some" instead of "some" is that as long as there is one, there are countless, and they differ from each other by any constant.

So the problem of proving that a force is conservative becomes the problem of finding a function.

We know, of course, that work independent of path leads to a corollary: work on any closed path is zero. But since there are an infinite number of closed paths, this inference cannot be proved either. Unless you want to use it to prove that a force isn't conservative.

One might say that if one proves that the curl of a force is zero, then it is conservative.

The reason why some people say this is because they are based on the experience that if the curl of a force is zero everywhere, then it is equivalent to proving that anywhere the integral of a force along an infinitely small closed path is zero, then the integral of any closed path obtained by adding them all is naturally zero!

In other words, they argue that a curl of zero everywhere proves that the integral of any closed path is zero, which means that the work is really independent of the path, and therefore the force must be conservative.

Is that right?

Not necessarily!

The reason is that the integrals of infinitesimal closed paths do not necessarily add up to the integrals of a closed path surrounding them.

As shown in the figure above, the shaded areas represent the distribution areas of the field.

In spaces of type L=0, the sum of the integrals of an infinite number of tiny loops must equal the integral of a large loop surrounding them. A force defined in this region is conservative if the curl is zero.

However, for L>0, since this condition is not satisfied, even if the rotation of the force is zero everywhere, it is not necessarily conservative.

But conversely, if it is a conservative force, then the curl of its force field must be zero everywhere, so the curl of zero everywhere is a necessary but not sufficient condition for conservative forces.

All right, now that we've talked about conservative forces, let's go back to conservative fields.

Its definition is exactly similar to conservative forces, namely:

Integrating path-independent vector fields. Note the qualifier-"vector field." This is not mentioned in the conservative force because the force itself is a vector.

Similarly, this definition is not practical. But similarly, if we can find some scalar function whose gradient (or negative gradient, to be exact) is the vector field, then it proves to be a conservative field.

Similarly, there are many ways to prove that a field is nonconservative.

The integrals of fields along different paths between two points can be compared. If they are different, they are non-conservative fields.

It is also possible to check that the integral of the field on a closed path is non-conservative if it is not zero.

A familiar example is that an electrostatic field is always a gradient of a negative potential function, so it is conservative.

The magnetic induction is not a conservative field, because the B-line is always closed, and it must not be zero as long as it is integrated along a B-line.

The key question is: what is the relationship between conservative fields and conservative forces? Only conservative fields can cause conservative forces? Nonconservative fields must lead to nonconservative forces?

A lot of people think so, but not necessarily.

Non-conservative fields can also lead to conservative forces, and there is no necessary connection between conservative fields and conservative forces.

A typical example is the force exerted on a magnetic moment in a magnetic field.

In the article "can magnetism really do infinite work? Where does its energy come from? In this paper, a simple method is used to give the law of magnetic moment acting force. After all, the work was originally a moment, and then it was directly changed into the result of extracting force from it. The process was indeed a bit slippery, but the result itself was not a problem.

Some people doubt the reason is: magnetic moment is a current ring, so there should be countless forces on itself, torque is the overall effect of these forces, you now directly replace a force, it seems unreasonable.

Doesn't that seem reasonable? But in fact this is a misunderstanding, magnetic moment is not a point!

Yes, the magnetic moment is the same as the point charge model. It is also a point model. Here is the field distribution of a magnetic moment. Since it is a point, the force on it is of course the only certainty.

In fact, when the current density and magnetic field force of a point in space are strictly calculated, this result will also be obtained, but the process is much more complicated. Any electrodynamics textbook has a detailed calculation process for the part related to static magnetic field energy.

The calculated results show that the magnetic moment force is a gradient of scalar function, which is conservative.

In short, although the intensity of magnetic induction itself is not a conservative field, it is amazing that its force on the magnetic moment is actually a conservative force.

For example, when there is no conduction current in space, the relationship between electric field, magnetic field and magnetic excitation vector is

Obviously, since there is no rotation at this time, in fact, its field lines are not closed, and can be regarded as a conservative field. The magnetic scale potential is proposed based on this.

But for magnetic induction, it is always closed, as shown in the figure above. It is evident that the curl of the magnetic induction is zero everywhere, both inside and outside the magnet, since there is no current density. However, since its loop integral still exists, it means that there are points on the surface of the magnet where the curl is not zero, corresponding to a current density, which is the magnetizing current.

Because magnetic induction is always inseparable from the charge or current in motion, there are many kinds of currents, including free current, magnetization current and polarization current (generally not mentioned in books) and displacement current. According to Ampere loop theorem, this determines that magnetic induction must be a closed curve, so it has no source and its divergence is always zero.

It's the simplest of Maxwell's equations.

For magnetic field strength, it can also be excited by moving electric charge or current. The part excited by current can be divided into two different types, the first is excited by free current, and the second is excited by other currents. This second magnetic field can be equivalently regarded as magnetically excited, an active field, and of course a conservative field.

Therefore, when there is no free current-for example, in the space of various magnetic media-the magnetic field strength is similar to the electrostatic field, and the laws used are exactly the same. However, since magnetic monopoles have not yet been discovered, this description is generally only an equivalent theoretical treatment.

At this point, you probably understand that the attraction and repulsion between magnets, if viewed from the point of view of magnetic field strength, resembles the action of a conservative field, the work it does comes from the potential energy of the system, which must be finite. Therefore, the work done by magnetic forces must also be finite.

So, you can probably see now why is the name "X field strength" similar to the electric field strength? Apart from historical reasons, it is also because they are really physical quantities of equal status.

As for its mathematical structure, it is equivalent to the electric displacement vector in the electric field, so its name is not so direct, but called "magnetic induction intensity."

But then again, physically speaking, sum is the basic physical quantity. Physical effects that are actually observable. Historically, there are certain irrationalities in naming, but there are always reasons for irrationality. There are also reasons for not changing it. Existence is reasonable!

In addition to and, vector potential and scalar potential are introduced to describe electromagnetic fields for mathematical convenience. Later, it was found that vector potential and scalar potential have observable quantum effects, so vector potential and scalar potential have become indispensable basic physical quantities to describe fields. They satisfy gauge transformation, which is a universal law in fundamental interactions.

This article comes from Weixin Official Accounts: University Physics (ID: wuliboke), by Xue Debao

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