Special relativity and magnetism

[BruceSpringste]

Hi!

I am trying to understand how magnetic fields actually are electric fields viewed in a different frame of reference. I understand the basics of it, however I am confused by how the magnetic field is rotational around a wire when current moves through it and why the force isn't "outwards" at all points.

Thanks!

 

[UltrafastPED]

Do you mean this: http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html

 

[jartsa]

Hi!

I am trying to understand how magneticfields actually electricfields viewed in a different frame of reference. I understand the basics of it, however I am confused by how the magnetic field is rotational when current moves through a wire and why the force isn't "outwards" at all points.

Thanks!

The force is "outwards". And "inwards".

A compass needle near a wire is pushed outwards and pulled inwards, which causes it to point to some direction, which is the direction of the magnetic field.

I mean, the microscopic magnets inside the compass needle are pulled and pushed, the same way as a loop of wire would be pulled and pushed.

Picture:
_____

O

Top part of the wire loop above is attracted towards the wire, if currents point the same direction in the wire and in the top part of the loop.

 

[BruceSpringste]

Do you mean this: http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html

That's exactly what I meant thanks!

The force is "outwards". And "inwards".

A compass needle near a wire is pushed outwards and pulled inwards, which causes it to point to some direction, which is the direction of the magnetic field.

I mean, the microscopic magnets inside the compass needle are pulled and pushed, the same way as a loop of wire would be pulled and pushed.

Picture:
_____

O

Top part of the wire loop above is attracted towards the wire, if currents point the same direction in the wire and in the top part of the loop.

That's what I don't understand. Shouldn't either the compass needle be pulled inwards or pushed outwards whatever orientation it has around the wire? Why does it follow a circular motion around the wire and not just outwards everywhere?

 

[ZapperZ]

I understand the basics of it, however I am confused by how the magnetic field is rotational around a wire when current moves through it and why the force isn't "outwards" at all points.

Thanks!

This really isn't "relativity" but rather straightforward basic E&M. The reason for the geometry of the field has a lot to do with the symmetry of the source.

Let's look at the situation when something has a field that points "outwards". The easiest example is an infinite uniform line charge. Look at the symmetry of the source. It has translational symmetry along the line charge, i.e. if the line is oriented in the z-axis, shifting the line charge up and down along the z-axis doesn't change anything. It also has rotational symmetry along a rotational axis in the z-direction. And, it also has a reflection symmetry, i.e. if I take the source and invert z -> -z, nothing has change.

If you go out of the room, and I perform any of these operations, you won't know the difference when you come back in and look at the charge distribution again.

The field that is generated must also reflect the same symmetry, i.e. if I perform the same operations, you also cannot tell that the field has change. Otherwise, you will have an inconsistency.

Now, go back to the magnetic field case, Say I have current going in the +z direction. Now, I have the same translational symmetry, and I have the same rotational symmetry of the source. However, if I do a "reflection" or inversion symmetry, i.e. I switch z -> -z, there is a difference, because now, the current is no longer moving in the +z direction, but rather in the -z direction. I can now see a "broken symmetry" here for this operation.

So the field that is generated must reflect the same characteristic as the source, which in this case, is the current flow. A concentric circular field having a particular sense of circular direction matches the same symmetry property of the current source. It has translational symmetry, and rotational symmetry, but if you flip the source 180 degrees, the circular direction flips into the opposite direction, just like what happened to the source!

This is why the magnetic field for a long straight wire has this geometry.

Zz.

 

[BruceSpringste]

This really isn't "relativity" but rather straightforward basic E&M. The reason for the geometry of the field has a lot to do with the symmetry of the source.

Let's look at the situation when something has a field that points "outwards". The easiest example is an infinite uniform line charge. Look at the symmetry of the source. It has translational symmetry along the line charge, i.e. if the line is oriented in the z-axis, shifting the line charge up and down along the z-axis doesn't change anything. It also has rotational symmetry along a rotational axis in the z-direction. And, it also has a reflection symmetry, i.e. if I take the source and invert z -> -z, nothing has change.

If you go out of the room, and I perform any of these operations, you won't know the difference when you come back in and look at the charge distribution again.

The field that is generated must also reflect the same symmetry, i.e. if I perform the same operations, you also cannot tell that the field has change. Otherwise, you will have an inconsistency.

Now, go back to the magnetic field case, Say I have current going in the +z direction. Now, I have the same translational symmetry, and I have the same rotational symmetry of the source. However, if I do a "reflection" or inversion symmetry, i.e. I switch z -> -z, there is a difference, because now, the current is no longer moving in the +z direction, but rather in the -z direction. I can now see a "broken symmetry" here for this operation.

So the field that is generated must reflect the same characteristic as the source, which in this case, is the current flow. A concentric circular field having a particular sense of circular direction matches the same symmetry property of the current source. It has translational symmetry, and rotational symmetry, but if you flip the source 180 degrees, the circular direction flips into the opposite direction, just like what happened to the source!

This is why the magnetic field for a long straight wire has this geometry.

Zz.

Now I understand. I thought it had the same logic as a moving charge explained in the comment above. But essentially the magnetic field is an electric field right? Or am I misunderstanding something? I only have a background of high school math so I might be a little slow at understanding this!

 

[ZapperZ]

But essentially the magnetic field is an electric field right?

It is not that simple. While you probably may not be able to comprehend this, I'll mention it anyway. The relationship between the magnetic field and electric field isn't a simple one, as described in the standard Maxwell equations. They are related by a mathematical operation that depends on the time rate of change of the field, and the "curl" or the circular nature of the field. So it depends on the change in time and the change in space of each of the field.

It isn't just "essentially ... an electric field".

Zz.

 

[BruceSpringste]

It is not that simple. While you probably may not be able to comprehend this, I'll mention it anyway. The relationship between the magnetic field and electric field isn't a simple one, as described in the standard Maxwell equations. They are related by a mathematical operation that depends on the time rate of change of the field, and the "curl" or the circular nature of the field. So it depends on the change in time and the change in space of each of the field.

It isn't just "essentially ... an electric field".

Zz.

With the possibility of being annoying; what is the magnetic field then?
Because from what you are saying I am under the assumption that the electric field is the magnetic field in another frame of reference aka "time rate of the field, and the curl" as you mention. However the magnetic field is perpendicular to the electric somehow. By influencing space and time the magnetic field will become an electric field. Or am I entirely wrong?

Maybe I am misunderstanding totally what this video is saying:

 

[ZapperZ]

With the possibility of being annoying; what is the magnetic field then?
Because from what you are saying I am under the assumption that the electric field is the magnetic field in another frame of reference aka "time rate of the field, and the curl" as you mention. However the magnetic field is perpendicular to the electric somehow. By influencing space and time the magnetic field will become an electric field. Or am I entirely wrong?

Well, what is electric field then?

The question "what is..?" is vague in physics. If you look closely, when I ask you "what is xxx?", you'll notice that the answer you can give is a series of description and properties of that xxx. If I ask you "what is an apple?", your answer will be "It is a fruit, it grows on trees, it can be red, green, yellow, or some combination of those colors, etc... etc...". In other words, you can only give me the set of characteristics that describes what an apple is.

So when you ask "what is a magnetic field?", I can only give you the set of properties of what it is. This is true for ANYTHING.

The electric field isn't more "fundamental" than magnetic field, and the magnetic field isn't more fundamental than the electric field. You cannot say one is simply the other, the same way you can't say that mass is nothing more than energy, even though one can be converted to the other.

So unless you can make a more definitive request on what you really want when you ask "What is a magnetic field?", I can only continue to give you all the physics that describes a magnetic field.

Zz.

 

[BruceSpringste]

Well, what is electric field then?

The question "what is..?" is vague in physics. If you look closely, when I ask you "what is xxx?", you'll notice that the answer you can give is a series of description and properties of that xxx. If I ask you "what is an apple?", your answer will be "It is a fruit, it grows on trees, it can be red, green, yellow, or some combination of those colors, etc... etc...". In other words, you can only give me the set of characteristics that describes what an apple is.

So when you ask "what is a magnetic field?", I can only give you the set of properties of what it is. This is true for ANYTHING.

The electric field isn't more "fundamental" than magnetic field, and the magnetic field isn't more fundamental than the electric field. You cannot say one is simply the other, the same way you can't say that mass is nothing more than energy, even though one can be converted to the other.

So unless you can make a more definitive request on what you really want when you ask "What is a magnetic field?", I can only continue to give you all the physics that describes a magnetic field.

Zz.

The electric field is made up of charges. What is the magnetic field made up of?

 

[WannabeNewton]

With the possibility of being annoying; what is the magnetic field then?

Both the electric and magnetic fields are but components of a single electromagnetic field. When we boost from one frame to another, both the electric and magnetic components of the electromagnetic field will in general mix with one another. This is what you are trying to allude to.

However this does not mean that the magnetic field is "just the electric field in another frame". The magnetic field is fundamentally different from the electric field. This can be easily seen from the definitions of the electric and magnetic fields in terms of the electromagnetic field.

 

[WannabeNewton]

The electric field is made up of charges.

The electric field is directly sourced by charges and the magnetic field is directly sourced by currents but they aren't made up of charges or currents. Note that a dynamical electric field does not need charges locally in order to exist. Maxwell's equations allow for the propagation of electromagnetic waves in vacuum wherein there are no local charges at all and the dynamical electric and magnetic fields generate one another through induction.

 

[BruceSpringste]

Both the electric and magnetic fields are but components of a single electromagnetic field. When we boost from one frame to another, both the electric and magnetic components of the electromagnetic field will in general mix with one another. This is what you are trying to allude to.

However this does not mean that the magnetic field is "just the electric field in another frame". The magnetic field is fundamentally different from the electric field. This can be easily seen from the definitions of the electric and magnetic fields in terms of the electromagnetic field.

I suppose this was the enough dumbed down version I was looking for thanks!

 

[Nugatory]

The electric field is made up of charges. What is the magnetic field made up of?

The electric field is not made up of charges. It is something that appears near charges.

 

[ZapperZ]

The electric field is made up of charges. What is the magnetic field made up of?

Magnetic field is made up of moving charges and/or the magnetic moment of angular and spin quantum moments.

This is a very vague "association" game that doesn't produce a logical explanation for your question.

Zz.

 

[jartsa]

That's exactly what I meant thanks!
That's what I don't understand. Shouldn't either the compass needle be pulled inwards or pushed outwards whatever orientation it has around the wire? Why does it follow a circular motion around the wire and not just outwards everywhere?

Do you want a compass needle that is pulled into a wire or pushed away from the wire?

Take a very thin wire and place the needle very close to the wire, that should work.

Or alternatively just make the needle an extremely easily moving one.
Picture1:________________
O

Here a wire loop is far from the straight wire, so the force pulling the top part of the loop is almost the same as the force pushing the lower part, because the top part and the lower part are almost at equal distance from the wire.

Picture2:________________

O

Here a wire loop is very close to a straight wire, so the force pulling the top part of the loop is larger than the force pushing the lower part. Loop moves towards the wire.

 

[UltrafastPED]

what is the magnetic field then?

Mathematically the electric field is a 1-form, while the magnetic field is a 2-form.

Physically, the electric field has a source: positive or negative electric charges
The magnetic field is _sourceless_: there are no magnetic charges - every magnetic has both a North and a South.

Maxwell's equations show how they interact. There is a geometric object - the Maxwell tensor - which represents both at the same time.

 

[BruceSpringste]

The answers here have been very helpful but I quickly realize that this might be a lot more complicated than I previously thought. In order to go in depth and understand more I need to study Maxwells equations?

 

[WannabeNewton]

Mathematically the electric field is a 1-form, while the magnetic field is a 2-form.

Both the electric and magnetic fields are vector fields. To every vector field there is an associated 1-form so both the electric and magnetic fields can be represented by 1-forms. But the magnetic field is certainly not a 2-form. The 2-form is the electromagnetic field. The magnetic field can be obtained from this 2-form by contracting with the projection of the natural volume element on space-time onto the local rest space of a given observer much like the angular velocity vector can be obtained from the exact same volume form projection contracted with the twist (rotation) 2-form.

 

[UltrafastPED]

Both the electric and magnetic fields are vector fields. To every vector field there is an associated 1-form so both the electric and magnetic fields can be represented by 1-forms. But the magnetic field is certainly not a 2-form. The 2-form is the electromagnetic field. The magnetic field can be obtained from this 2-form by contracting with the projection of the natural volume element on space-time onto the local rest space of a given observer much like the angular velocity vector can be obtained from the exact same volume form projection contracted with the twist (rotation) 2-form.

You are correct; I should have checked a reference prior to posting.

 

[TrickyDicky]

In fact one can mathematically define the magnetic field as a 2-form if one stays in the original three dimensional Euclidean EM formulation by Maxwell. The magnetic field is considered a pseudovector which is simply the Hodge dual equivalent of a 2-form in the specific Euclidean 3D setting. So you were correct UltrafastPED.