# How does magnetism occur?

1. Aug 16, 2006

### pivoxa15

Magnetic fields arise because of moving charges which is the existence of a current (given by Ampere's law).
Alternatively, a magnetic dipole is set up as a result of a charge moving around in a circular loop. So it follows that magnetism occurs from moving charges moving in different ways.

But people say that magnetism arise because of special relativity. How does SR come into it since it dosen't matter how fast the charges are moving?

Last edited: Aug 16, 2006
2. Aug 16, 2006

### rbj

plus the electrostatic effects of charges.

your premise in this last question is wrong. where did you get that?

3. Aug 17, 2006

### pivoxa15

your premise in this last question is wrong. where did you get
that?[/QUOTE]

I just assumed this. I know light must travel at speed of light but charges which are electrons can't reach that speed. Plus most of the time they don't travel at those speeds unless when they are in a particle accelerator. I certainly wouldn't expect electrons in my computers to be at speed of light or would I?

Mass movement of electrons at 0.001speed of light would also constitute a current hence generate a magnetic field somewhere wouldn't it?

4. Aug 17, 2006

### dextercioby

Special relativity comes in connection to the em tensor, not with the relevance of the speed of charged particles...

Daniel.

5. Aug 17, 2006

### cesiumfrog

It can be explained quite simply, see Griffith's "Electrodynamics" for example.

Think of any situation, say a test-charge moving beside a current-carrying wire, and use special relativity to consider the situation from the test-charge's frame of reference. In this frame (where the test-charge is stationary for the moment) the electron-current in the wire is moving at a different velocity than the wire itself (which constitutes a current of positive charged atoms). Special relativity says that the Lorentz-contraction of the spacing between the current electrons is therefore different to the contracted spacing between positive atoms in the wire. Since the densities of positive and negative charges are hence different in the test-charge's frame of reference, there is an electrostatic force on the test-charge.

When you look at such situations from a different frame of reference (eg. from the point of view where both the electron-current-velocity is equal and opposite to the velocity of the atoms composing the wire, so that positive and negative charge densities are in balance), the force you just calculated for the test-charge can no longer be explained electrostatically: In this frame of reference, it is called the "magnetic" force (although really it just arose from SR).

Since this arose through relativistic Lorent-contractions, it does depend how fast the charges are moving.. just as the conventional laws of electromagnetism do (eg. explicitly in the Lorentz-force law, and implicitly in the definition of currents).

6. Aug 17, 2006

### Staff: Mentor

Moving charges induce a magnetic field, regardless of speed.

Electrons do not move at the speed of light, but the electric field generated by the charge does in a sense.

Maxwell's equations can be derived from SR, to which dextercioby alluded.

7. Aug 18, 2006

### rbj

yeah, but, for a fixed charge, the strength of the magnetic field does depend on the speed.

i can't do tensors, so my understanding of the math used in SR is limited to sophmore/junior introductory modern physics. but this thought experiment (which is not general, only an illustrative example) persuaded me that magnetic fields (from a classical physics POV) come from the sole electrostatic field with SR taken into account:

The simplest hypothetical experiment would be two identical parallel infinite lines of charge (with charge per unit length of $\lambda$ ) and some non-zero mass per unit length of $\rho$ separated by some distance $R$. If the lineal mass density is small enough that gravitational forces can be neglected in comparison to the electrostatic forces, the static non-relativistic repulsive (outward) acceleration (at the instance of time that the lines of charge are separated by distance $R$) for each infinite parallel line of charge would be:

$$a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}$$

If the lines of charge are moving together past the observer at some velocity, $v$, the non-relativistic electrostatic force would appear to be unchanged and this would be the acceleration that an observer traveling along with the lines of charge would observe.

Now, if special relativity is considered, the clock of the observer moving along with the lines of charge would be ticking at a relative rate (ticks per unit time or 1/time) of $\sqrt{1 - v^2/c^2}$ from the point-of-view of the stationary observer because of time dilation. Since acceleration is proportional to (1/time)2, the at-rest observer would observe an acceleration scaled by the square of that rate, or by ${1 - v^2/c^2}$, compared to what the moving observer sees. Then the observed outward acceleration of the two infinite lines as viewed by the stationary observer would be:

$$a = \left(1 - v^2 / c^2 \right) \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}$$

or

$$a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} - \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho} = \frac{ F_e - F_m }{\rho}$$

The first term in the numerator, $F_e$, is the electrostatic force (per unit length) outward and is reduced by the second term, $F_m$, which with a little manipulation, can be shown to be the classical magnetic force between two lines of charge (or conductors).

The electric current, $i_0$, in each conductor is

$$i_0 = v \lambda$$

and $$\frac{1}{\epsilon_0 c^2}$$ is the magnetic permeability

$$\mu_0 = \frac{1}{\epsilon_0 c^2}$$

because $$c^2 = \frac{1}{ \mu_0 \epsilon_0 }$$

so you get for the 2nd force term:

$$F_m = \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} = \frac{\mu_0}{4 \pi} \frac{2 i_0^2}{R}$$

which is precisely what the classical E&M textbooks say is the magnetic force (per unit length) between two parallel conductors, separated by $R$, with identical current $i_0$.

so you can look at it two ways: one is purely classical where the magnetic force exists and has a separate origin from the electrostatic force and effects of relativity are not considered. the other is where there is only the electrostatic force but the effects if special relativity are considered. both results appear the same to the "stationary" observer.

8. Aug 18, 2006

### rbj

no matter what i do, i can't get LaTeX equations to render. what is wrong?

9. Aug 18, 2006

### Staff: Mentor

I posted some very simple LaTeX a little while ago, which failed with the same mesages. I'm 99.9% sure it's just a temporary glitch in the forum software. I won't be surprised if our stuff actually appears correctly sometime tomorrow (oops, later today).

10. Aug 20, 2006

### rbj

geez. dunno how temporary this glitch is. i tried it again with a new post (and deleted it when it didn't work).

site down for 2+ days and this problem. i hope they figure it out.

11. Aug 20, 2006

### rbj

now that LaTeX is working, a repost.

yeah, but, for a fixed charge, the strength of the magnetic field does depend on the speed.

i can't do tensors, so my understanding of the math used in SR is limited to sophmore/junior introductory modern physics. but this thought experiment (which is not general, only an illustrative example) persuaded me that magnetic fields (from a classical physics POV) come from the sole electrostatic field with SR taken into account:

The simplest hypothetical experiment would be two identical parallel infinite lines of charge (with charge per unit length of $\lambda$ ) and some non-zero mass per unit length of $\rho$ separated by some distance $R$. If the lineal mass density is small enough that gravitational forces can be neglected in comparison to the electrostatic forces, the static non-relativistic repulsive (outward) acceleration (at the instance of time that the lines of charge are separated by distance $R$) for each infinite parallel line of charge would be:

$$a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}$$

If the lines of charge are moving together past the observer at some velocity, $v$, the non-relativistic electrostatic force would appear to be unchanged and this would be the acceleration that an observer traveling along with the lines of charge would observe.

Now, if special relativity is considered, the clock of the observer moving along with the lines of charge would be ticking at a relative rate (ticks per unit time or 1/time) of $\sqrt{1 - v^2/c^2}$ from the point-of-view of the stationary observer because of time dilation. Since acceleration is proportional to (1/time)2, the at-rest observer would observe an acceleration scaled by the square of that rate, or by ${1 - v^2/c^2}$, compared to what the moving observer sees. Then the observed outward acceleration of the two infinite lines as viewed by the stationary observer would be:

$$a = \left(1 - v^2 / c^2 \right) \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}$$

or

$$a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} - \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho} = \frac{ F_e - F_m }{\rho}$$

The first term in the numerator, $F_e$, is the electrostatic force (per unit length) outward and is reduced by the second term, $F_m$, which with a little manipulation, can be shown to be the classical magnetic force between two lines of charge (or conductors).

The electric current, $i_0$, in each conductor is

$$i_0 = v \lambda$$

and $$\frac{1}{\epsilon_0 c^2}$$ is the magnetic permeability

$$\mu_0 = \frac{1}{\epsilon_0 c^2}$$

because $$c^2 = \frac{1}{ \mu_0 \epsilon_0 }$$

so you get for the 2nd force term:

$$F_m = \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} = \frac{\mu_0}{4 \pi} \frac{2 i_0^2}{R}$$

which is precisely what the classical E&M textbooks say is the magnetic force (per unit length) between two parallel conductors, separated by $R$, with identical current $i_0$.

so you can look at it two ways: one is purely classical where the magnetic force exists and has a separate origin from the electrostatic force and effects of relativity are not considered. the other is where there is only the electrostatic force but the effects of special relativity are considered. both results appear the same to the "stationary" observer.

Last edited: Aug 21, 2006
12. Aug 21, 2006

### pivoxa15

I haven't read the above derivation in detail. But I think I have a better understanding of why magnetism just is electricity or vice versa, depending on one's reference frame.

But I have a qualitative question.

With the example of two infinite paralle wires carrying the same (positive charge) current in the same direction, there is an inward attraction between the two wires. Now pretend we sit on a charge and move with the current. There must still be an inward force attracting me to the opposite wire but now I must use my electrostatic equations to calculate this force since I see the positive charge opposite me as stationary and no magnetic field exists according to me. The problem is in order to be attracted to this stationary charge, that charge must turn into a negative charge or the charge I am sitting on has turned negative. But how can this happen! Why is it that when I change my frame of reference, the charge on one wire must change? That seems absurd. What is wrong?

13. Aug 21, 2006

### WhyIsItSo

It is working now, but seems to not affect previously created LaTeX. If it is your message, return to it, edit it, and make a small change to the LaTeX, such as add a space (which will be ignored) which will trigger the system to recreate the image. Has worked for me, except for a few posts for which the edit option is not available.

14. Aug 21, 2006

### rbj

it is if you consider the effects of relativity.

from the POV of SR, no inward attraction, just a reduction of apparent outward repulsion that is due to time dilation.

it doesn't happen that way. first of all, you, the observer, aren't charged nor attracted to either line of charge. from your POV, you are simply sitting beside two identical parallels lines of charge that are repelling each other. as you fly by the other "stationary" observer, he looks at your clock and sees that it is ticking slower than his clock. that means the rate that he sees the two lines of charge repelling each other is slower than the rate you see them repelling each other.

now, for the stationary observer, if you do not consider the effect of SR, then what explains the reduced rate of repulsive acceleration between the two lines of charge? in classical physics, we would say that there are two forces acting on the lines of charge. one is the electrostatic repulsion which would be the same whether the lines are moving or not and the other would be an electromagnetic attraction that is quantitatively less than the repulsion until the speed of the lines whizzing past the stationary observer reaches the speed of light.

in that case, the magnetic attraction is just equal to the electrostatic repulsion according to classical physics. according to SR, the clock of the moving observer (you), has come to a stop from the POV of the stationary observer. of course, those lines can't move that fast from anyone's POV.

Last edited: Aug 21, 2006
15. Aug 22, 2006

### pivoxa15

Are you assuming the wire is net positively charged or neutral? If the former then you are saying that the attractive force (from the magnetic field) which we usually calculate without considering SR, is actually a force that is meant to reduce the force of repulsion between the two wires with currents?

I have seen experiments where when a current is turned on in two conductors, they move toward each other (i.e. aluminium foil). This would be the case where the two conductors have the same charges and so instead of repelling less, they attract?

16. Aug 22, 2006

### rbj

i'm saying the former (but i'm not calling it a "wire" but a "line of charge"). also, the lines of charge don't have to be both positive, just identically charged so both positive or both negative, with the same amount of charge distributed per unit length. so, all by themselves, these two infinite and parallel lines of charge will repel. but from the POV of classical E&M, their repulsion is reduced by an attractive magnetic force if they are moving. in classical E&M, the one moving line sets up a magnetic field depicted as concentric circles going around the line of charge. and the other line moves perpendicularly through the magnetic field and thus has a magnetic force acting on it. this magnetic force acts along with the electrostatic force but in the opposite direction. and if $v < c$, it is never as large as the repulsive electrostatic force, so the net force is still outward, but reduced.

in the SR POV, there is no magnetic force, there is only electrostatic, but because of time dilation, the lines appear to the "stationary" observer to be moving apart at a reduced acceleration compared to if they weren't moving past the observer.

it is because these are "wires" not hypothetical "lines of charge". the hypothetical lines of charge are not neutral, they have only one type of charge and it is all moving (so there is a current). the wires are electrically neutral. the electrons are moving and the protons in the nuclei of the atoms are not moving. the electrostatic repulsion that the like charges of the two lines experience is canceled by the electrostatic attraction the unlike charges have for each other and, because of atomic phenomena, even though the electrons are free to move along the wires, they can't jump off, so the attractive forces and the repulsive forces are "mechanically" coupled and the wires don't move (because of electrostatics) because of these balanced electrostatic forces.

but if the electrons are made to move for some reason, then there are magnetic forces remaining that aren't cancelled. now how to look at this from the SR POV (leaving out the magnetic force but keeping the electrostatic force), i am not sure how this would be done because the "moving" observer (moving along with the electrons) would see the electrons as stationary but see the protons moving in the opposite direction. somehow both observers would have to see the lines or wires moving toward each other at some rate and not repelled. but i am not sure how to set it up. that's why i left the thought experiment as two lines of charge and not neutral wires with both positive and negative charges. it's easier to get a handle on that and it does offer some explaination of the phenomenon but there is both electrostatic and electomagnetic forces from the classical POV (and electrostatic from the SR POV).

Last edited: Aug 22, 2006
17. Aug 23, 2006

### pivoxa15

The book explains this phenomena like this: When my reference frame is stationary on the moving charge, the charge distribution I see is different to when I saw the charge distribution from another inertial perspective. My fixed perspective led me to see equal charges on the wires so the inward attraction is the magnetic force. But when my reference frame was moving at constant speed relative to the wire, I see the charge distribution change in the wire due to Lorentz constrations and hence more negative charges in the wire than positive since the negatives are moving and the positives including the one I am stitting on is staionary. Although I haven't told the details welll. This leads to me being attracted to the wire which to me is net negative.

18. Aug 23, 2006

### rbj

i think i can buy that.