# A Couple Magnetism Questions

• I
Hello, there are a couple things about magnetism that I do not understand.

1. Why didn't we define the magnetic field to be in the directions of the force? This isn't really a technical question, I am just more curious about why it is this way. The way I was thinking of it, the math seems to get a little more complicated. The velocity would be perpendicular to the magnetic field if it were defined that way, but we can't really represent this with the cross product because the direction would be wrong. So is it just for lack of a better mathematical representation?

2. How does magnetism fit in with relativity? Not really advanced special relativity, but just the fact that all motion is relative. From my understanding of it, a magnetic field is caused by a moving source charge (or collection of charges), but a charge (not the source charge) only feels a force if it is moving. So, if we change our reference frame to make the source charge stationary, wouldn't the source charge only cause an electric field? There must be some flaw in my understanding of magnetism here because this would mean that a motionless charge can feel a magnetic force.

## Answers and Replies

Dale
Mentor
2020 Award
Why didn't we define the magnetic field to be in the directions of the force?
Because the direction of the force depends not just on the field but also on the velocity of the charge. Thus the same field can have a force in almost any direction.

So, if we change our reference frame to make the source charge stationary, wouldn't the source charge only cause an electric field? There must be some flaw in my understanding of magnetism here because this would mean that a motionless charge can feel a magnetic force.
This is correct. The motionless charge does not feel a magnetic force, but instead what is considered an electric force and what is considered a magnetic force depends on the reference frame. The electromagnetic force can be part magnetic and part electric in one frame and all electric or all magnetic in another frame, depending on the details.

Okay, that is interesting. I guess the next question I have is what would happen if two people are observing it in different reference frames? For example, imagine a motionless source charge in one frame with a charge moving by it quickly to the left. If they are of opposite charge, the moving charge would be attracted to the source charge slightly and would kind of spiral around it a little bit. Now, imagine a reference frame in which the charge is still moving left, but slower than in the first case and the source charge is now moving to the right. So if we use the right hand rule for the current moving to the left we get a magnetic field, and using the right hand rule again to cross the velocity with the magnetic field, I get that the direction is away from the source charge, so it would be pushed away from it.

Actually, as I just finished writing this I may have figured it out. The charges are opposite and I forgot to account for this when considering the force. In the second case, the force would be in the opposite direction I determined and would therefore exhibit the same behavior as in the first reference frame. Is my reasoning here all correct?

Mister T
Science Advisor
Gold Member
Why didn't we define the magnetic field to be in the directions of the force?

Which force? A little context would help. Say you have a proton moving in the x-direction. A magnetic force is exerted on the proton, that force is in the y-direction. Can you tell me which way the needle of a compass would point, assuming the presence of the proton has a negligible effect?

How does magnetism fit in with relativity?

As you reasoned, a magnetic field in one frame of reference can be an electric field in another.

Dale
Mentor
2020 Award
Is my reasoning here all correct?
Yes, it is worth going through the exercise of calculating the force.

A slightly easier scenario is to calculate the force between two charges at rest and then repeat the calculation from a frame where they are both moving with velocity v.

Which force? A little context would help. Say you have a proton moving in the x-direction. A magnetic force is exerted on the proton, that force is in the y-direction. Can you tell me which way the needle of a compass would point, assuming the presence of the proton has a negligible effect?

I meant the magnetic force. I was just wondering if it were possible to define the magnetic field in the direction of the magnetic force while maintaining mathematical simplicity.

I would think that the compass would point in the positive y-direction because the magnetic force would push the north end of the compass needle up since the source charge is positive.

Mister T
Science Advisor
Gold Member
I would think that the compass would point in the positive y-direction because the magnetic force would push the north end of the compass needle up since the source charge is positive.

No, the compass will point in a direction that is not in the xy-plane. So, if we define the direction of the magnetic field to be the same as the direction of the magnetic force, we will need to come up with some scheme that allows us to predict the direction the compass needle points.

The fact of the matter is that one must consider directions that are not all in the same plane. Never before in the college-level introductory courses was a phenomenon encountered that required this. It has to do with the way Nature behaves, not with way we define the direction of a magnetic field.

No, the compass will point in a direction that is not in the xy-plane. So, if we define the direction of the magnetic field to be the same as the direction of the magnetic force, we will need to come up with some scheme that allows us to predict the direction the compass needle points.

I still feel as though I'm missing something here. What is causing the compass to point in the direction it is? It should be a force of some kind, but it's not in the direction of an electromagnetic force. Perhaps it is possible that I have reached the most fundamental understanding of it that I need and asking what is causing it doesn't really have an answer, but my understanding still feels incomplete. I guess I am having trouble discerning what I should accept and what I need to understand in a more fundamental way.

Mister T
Science Advisor
Gold Member
I still feel as though I'm missing something here. What is causing the compass to point in the direction it is?

Why was this question not asked about the force exerted on the proton? Anyway, whatever it is you invent to explain the force on the proton can be used to explain the forces on the compass needle. They have the same cause.

It should be a force of some kind, but it's not in the direction of an electromagnetic force.

The forces exerted on the compass needle are not in the same direction as the force exerted on the proton. This is something we observe happening. The physics is an explanation. The force on the proton equals ##q\vec{v}\times\vec{B}##. The torque on the compass needle equals ##\vec{\mu}\times\vec{B}##.

I guess I am having trouble discerning what I should accept and what I need to understand in a more fundamental way.

The only thing you need to accept is the way Nature behaves. Physicists have explained this particular behavior using a magnetic field. I thought that was what you were trying to understand. The fact that the magnetic field and the force exerted on a moving charged particle in that field have different directions. The reason is because it describes what we observe happening in Nature.

Force in an electromagnetic field is something I was considering for giving an impulse to a pendulum; the issue being one of trying to minimise escapement error by zeroing the force around the bottom of the swing.
My solution to calculating force was to calculate the rate of change of energy (stored in a changing air gap and a changing magnetic flux) with changing pendulum position; based on energy = force x distance therefore force = dEnergy/dDistance
I don't know if this engineering approach helps ....

davenn
Science Advisor
Gold Member
Force in an electromagnetic field is something I was considering for giving an impulse to a pendulum; the issue being one of trying to minimise escapement error by zeroing the force around the bottom of the swing.
My solution to calculating force was to calculate the rate of change of energy (stored in a changing air gap and a changing magnetic flux) with changing pendulum position; based on energy = force x distance therefore force = dEnergy/dDistance
I don't know if this engineering approach helps ....

This thread isn't about EM fields. It's about static magnetic fields davenn
Science Advisor
Gold Member
What is causing the compass to point in the direction it is? It should be a force of some kind, but it's not in the direction of an electromagnetic force.

The compass needle, which is a magnet, it aligning itself with the external magnetic field
eg ... the magnetic field of the earth, or say, of a bar magnet you bring close to the compass

the South pole of the compass needle will be attracted to the North pole of the external field and visa versa for the other pole

The Lorentz force acting on an electric charge or a magnetic pole is given by
$$\vec{F}_m=p\vec{B}-(pc^{-2})\vec{v}\times\vec{E}$$
$$\vec{F}_q=q\vec{E}+q\vec{v}\times\vec{B}$$
respectively, where ##\vec{v}## is the velocity of the charge ##q## or the pole ##p##, and ##\vec{E}## and ##\vec{B}## are electric and magnetic fields, respectively, set up by other things. These equations are independent of Maxwell's equations, which give ##\vec{E}## and ##\vec{B}## given a charge and/or pole distribution. I find it very useful to define ##\vec{B}## using the first equation rather than the second, even though the second is the conventional one that is used to define ##\vec{B}## (Lorentz force acting on a moving charge). Using the first (magnetic field is the force acting on a magnetic pole) is far more intuitive and practical. I believe these are axially symmetric as they should be, but I'm not sure if they are special-relatively invariant.