Why moving charges create magnetic field?

[kent davidge]

(Sorry my poor English) A changing electric field creates a magnetic field and vice-versa. I thought one field were created only when the another field were disturbed and it works finely for electromagnetic waves, and it agrees with Maxwell's equations. I was happy with my conclusion then I found a problem: a moving charge actually creates a magnetic field EVEN with constant speed, it breakes my conclusion apart, because in this case the electric field is not being disturbed and yet we have a magnetic field. How could it be?

 

[EmilyRuck]

(Sorry my poor English) A changing electric field creates a magnetic field and vice-versa.

A time-varying electric field creates a magnetic field which is time-varying itself.

I thought one field were created only when the another field were disturbed and it works finely for electromagnetic waves, and it agrees with Maxwell's equations.

What you are saying is valid only for fields that vary with time. But not all the electric and the magnetic fields vary with time. When they are fixed in time, they are called static. For static fields, we can no more talk about "disturbance". Static electric and magnetic fields exist by their own. They are two different entities (mathematically represented by a vector field). The electro-static field can exist regardless of the static magnetic field and vice versa.

I was happy with my conclusion then I found a problem: a moving charge actually creates a magnetic field EVEN with constant speed, it breakes my conclusion apart, because in this case the electric field is not being disturbed and yet we have a magnetic field. How could it be?

A moving charge with constant speed creates a static magnetic field.
Maybe other users can add something more interesting about the nature of the electro-static field and the magneto-static field: what exactly are they? To convince that they are two different entities, as a fist suggestion, I could encourage you to observe the different effects that the two types of fields provoke. The electro-static field acts on electric charges, by moving there. The magneto-static field only acts on moving electric charges (even when the field is static, so even when it does not vary with time). If an electric charge is fixed in space (it does not move), even when a static magnetic field is established, the charge will not be subjected to it.

 

[ZapperZ]

A time-varying electric field creates a magnetic field which is time-varying itself.

Actually, this is not correct.

From Ampere's law, the curl of B is proportional to the time rate of change of E (and current density if there's one). But this curl of B need not have a time varying solution as well. It can easily be a magnetostatic field.

Zz.

 

[andresB]

The origin of electric and magnetic field are charges and currents as can be seen from Jefimenko's equations. On the other hand, besides "because it works", it is hard to answer "why" equations in physics.

 

[Orodruin]

A moving charge with constant speed creates a static magnetic field.

No it doesn't. You cannot have a static solution unless the source is static.

 

[kent davidge]

Oh okay. Let me try a different approach. How it's possible that an observer at the same speed of two charged particles doesn't observe a magnetic interaction between them, and an observer in another frame actually observes it?

 

[ZapperZ]

Oh okay. Let me try a different approach. How it's possible that an observer at the same speed of two charged particles doesn't observe a magnetic interaction between them, and an observer in another frame actually observes it?

Because in one frame, the observer detects a time-varying electric field, while in the other, it is an electrostatic field.

Zz.

 

[kent davidge]

But how it's possible that at the same time the same two particles are, say, running away each other as seen in one frame, and doesn't do that from the another frame?
sorry my english.

 

[Hornbein]

(Sorry my poor English) A changing electric field creates a magnetic field and vice-versa. I thought one field were created only when the another field were disturbed and it works finely for electromagnetic waves, and it agrees with Maxwell's equations. I was happy with my conclusion then I found a problem: a moving charge actually creates a magnetic field EVEN with constant speed, it breakes my conclusion apart, because in this case the electric field is not being disturbed and yet we have a magnetic field. How could it be?

The magnetic field is essentially a correction to the electric field due relativity. We assume no relativity, get the electric field, and use the magnetic field to correct for the difference. It works amazingly well.

 

[Hornbein]

But how it's possible that at the same time the same two particles are, say, running away each other as seen in one frame, and doesn't do that from the another frame?
sorry my english.

Right. The correction for relativity is different in the two cases. So the results for electric and magnetic field are also different.

 

[Orodruin]

Put in a different way: In relativity the electric and magnetic fields are really two sides of the same coin. The question is similar to asking why (in two dimensions) you will observe a force in the y-direction after a rotation when the before the rotation the force was purely in the x-direction.

 

[jtbell]

But how it's possible that at the same time the same two particles are, say, running away each other as seen in one frame, and doesn't do that from the another frame?

Can you give an example where this actually happens?

 

[kent davidge]

Can you give an example where this actually happens?

Well, let's say there are an observer in a laboratory, and another observer in a street outside. Now, we accelerate the laboratory (LOL) from t₀ to t₁. From the point of view of the observer in the lab, there was no moviment of charges and at t₁ they are in the same place as before. But from point of view of the observer outside the lab, the charges are now distant each other. If this observer move to laboratory, he would disagree with the other observer about what they seen.

 

[jtbell]

But from point of view of the observer outside the lab, the charges are now distant each other.

Why?

 

[kent davidge]

Why?

Because when the lab is moving relative to the street, a magnetic force acts on the charges.

 

[jtbell]

But an electric force also acts on the charges. The net electromagnetic force is always either attraction for "unlike" charges, or repulsion for "like" charges, regardless of the relative velocity of the observer.

 

[kent davidge]

But an electric force also acts on the charges

both observers will "see" the electric force acting on the charges. The thing is that for the observer at rest relative to the charges, there would be a electric force AND a magnetic force, whose action would be put the charges away each other. The another observer who travels at the same speed as the charges, would "see" only the effect of the electric force, therefore he would "see" the charges more closely. Could you understand me?

 

[Orodruin]

both observers will "see" the electric force acting on the charges. The thing is that for the observer at rest relative to the charges, there would be a electric force AND a magnetic force, whose action would be put the charges away each other. The another observer who travels at the same speed as the charges, would "see" only the effect of the electric force, therefore he would "see" the charges more closely. Could you understand me?

No, you are wrong. The electric field of a moving charge is not the same as that of a stationary one!

 

[jtbell]

Yes, the "stationary" observer "sees" only an electric force, and the "moving" observer "sees" both electric and magnetic forces. The electric forces "seen" by the two observers are not the same, just as the magnetic forces "seen" by the two observers are not the same. Indeed, even the net force is different as "seen" by the two observers! Force transforms under a Lorentz transformation as part of the four-force.

Nevertheless, it turns out that either all observers "see" a net attractive force, or they all "see" a net repulsive force.

 

[kent davidge]

Because in one frame, the observer detects a time-varying electric field, while in the other, it is an electrostatic field.

but if the electric field varies, then the magnetic field would also vary, right? Then how could the magnetic force be constant?

Nevertheless, it turns out that either all observers "see" a net attractive force, or they all "see" a net repulsive force.

But why the displacement of the charges would be the same as measured for each observer? When solving problems, we should add the magnetic and electric forces, like q.E + q.V.B. Each charge would have a greater acceleration [q(E+VB)/m] as measured by the "moving" observer than a acceleration q.E/m as measured by the" stationary" observer.

No, you are wrong. The electric field of a moving charge is not the same as that of a stationary one!

I remember when solving electrodynamics problems that a charge immersed on a electric field experiences the same F = q.E force as they experiences when at rest. Is there a equation for calculating the electric field of a moving charge? Why should we not consider the change in the electric field in that cases?

 

[kent davidge]

If the force acting on two charges as seen in a "stationary" frame is given by F_electric(1 - v²/c²), and since the acceleration of each charge is the same as measured in any frame of reference, then in order to law F = m.a be valid, the mass of each charge measured in the "stationary" frame must be greater than their mass in the "accelerated" frame. Is it so?

 

[Dale]

@kent davidge
I would strongly recommend trying to understand this in one frame first, before trying to understand it in multiple frames.

In a frame where one charge is moving at a relative velocity of you look at any point in space you will see that the E field is varying in time. This implies that there is a B field with a curl in space.

If you wish to calculate those fields, then the easiest approach will be Jefimenko's equations rather than Maxwells.

 

[kent davidge]

@Dale
I wish to understand what really is meant by "a field varying in time". Do we need a position in space to compare the intensity of the field in some time interval? It seems that the field of a charge with constant speed will vary with time on each point in space. Is it right? Some users mentioned in previous post that a observer in a frame with different speed will observe the field of a charge as disturbed, but I think they forgot the fact that I'm asking for a constant speed situation. In such situation all points on the field have the same speed, so is the field varying?

 

[Dale]

I wish to understand what really is meant by "a field varying in time".

It means $$\frac{\partial}{\partial t} E(t,x,y,z)\ne 0$$

Do we need a position in space to compare the intensity of the field in some time interval?

Yes. The partial derivative operator holds the other variables constant.

It seems that the field of a charge with constant speed will vary with time on each point in space. Is it right?

Yes.

In such situation all points on the field have the same speed, so is the field varying?

The field doesn't have a speed.

 

[kent davidge]

The field doesn't have a speed.

But suppose a charge moving from s₀ to s₁, its field should also move, so why it doesn't have speed?

 

[Dale]

But suppose a charge moving from s₀ to s₁, its field should also move, so why it doesn't have speed?

The charge has a speed, the field does not. Fields are not rigid globs of goo that surround particles.

 

[kent davidge]

@ Dale
Oh ok. Can you show me how to derive from Maxwell's equations a expression which relates the acceleration of a charge with the change in EM field?

 

[Dale]

Sure. For a point charge you should use the Lienard Wiechert potentials. For an arbitrary charge and field distribution you can use Jefimenkos equations.

https://en.m.wikipedia.org/wiki/Liénard–Wiechert_potential

https://en.m.wikipedia.org/wiki/Jefimenko's_equations

 

[kent davidge]

Thanks!