Electric Fields vs. Magnetism (Part 1)
You ask any physicist the definition of a magnetic field and he or she will either give you a textbook answer and or a mathematical answer. However, both are actually arbitrary. This is a little bit of a read but I urge you to read through all of it if you are going to make any comments or any attempt to criticize my thoughts.
I’ll repeat, I’m in the process of rendering the entirety of the subject of magnetism an arbitrary value. Does this mean people will stop teaching it? Absolutely not, because it’s so much more simplified (because of the fact it’s arbitrary) it will still go on being taught the same way; however, the goal here is to actually provide fundamental meaning of magnetism. What I mean by fundamental meaning to magnetism is prove that magnetism isn’t really anything but a simplified version of a very complex system of charges and electric fields.
My goal is to graphically, mathematically, and conceptually give reasoning for my theory in this blog; it may be a little disorganized but I have yet to actually organize it on paper yet so I have to start somewhere. In this part, part one of my explanations, a deep mathematical understanding of relativity isn’t necessary, just an abstract conceptual one.
I’m going to Photoshop an example diagram regarding my situation because that would look better than my MS Paint or blackboard drawings. It’s fairly abstract; however it’s also fundamental, which is what we’re aiming at. There are some obvious flaws but consider these findings in pre-beta version mathematically; conceptually it all makes sense, however much of it is a little counter-intuitive but will be explained later in some detail.
My labeling will seem counter-intuitive but when I get into the relativistic integration, it will make sense because gamma will need to be introduced sooner or later to make the correlation.
Picture two electrons, as I have, separated by distance ‘r’, where the second electron is traveling at velocity vector, in some arbitrary direction we will note as negative. Theta represents the angle between vector v and vector r. This isn’t too necessary but is required to be established here so it will make sense in the math later. Also, I’m restating the fact that I’m using relativity to define this therefore the fact that electrons have wave particle duality can be ignored because Euclidean geometry and Einsteinian mechanics work perfectly.
Think back a ways and remember what current is, aside from Ohm’s law, but on a more fundamental level. It is measured in amperes and is noted as the time derivative of charge (I’m going to do what everyone else does and label current, I, and charge, q): (try to picture this, it’s important because I make some nasty assumptions later). Well it’s also common sense that velocity is just the time derivative of a distance, r, . The definition of electric field, E, is the amount of force distributed for a certain charge and can be mathematically written as such: . Mathematically, it’s obvious that there is a direct correlation between electric field and current by the time derivative of charge. It can be written as follows: . We can rewrite the initial equation as therefore allowing us to establish an understanding of force which is how I will prove everything down the road.
Whenever there is the presence of a charge, there is an electric field that surrounds it and that electric field exerts a force on any surrounding charge proportional to the magnitude of something like 1/r^2. It’s 1/r^2 because we live in a three dimensional universe and everything expands outward from a center control with a radius. The only thing in three dimensions that corresponds to a radius that has equidistant unit points at every point in space, infinitesimally small units apart, is a sphere. Why not 1/r^3 if we’re in three dimensions? Think of the surface area of the sphere, it goes something like 4pi*r^2. When I get further into it, all the electrical constants have 4pi embedded into them somewhere, such as mu nought, coloumb’s constant, epsilon nought, etc. That’s where the r^2 comes from and it’s a very important concept to grasp. This is why magnetic field can’t be specifically described as “real” in the common sense because it goes by a proportional relationship like 1/r, not 1/r^2.
Taking this from Google’s definition feed, “magnetism is one of the phenomena by which materials exert an attractive or repulsive force on other materials”. Magnetism exists from the movement of electrical charge. Remember what I said earlier, the movement of electrical charge in a quantitative format is described as current. This needs to be defined on a relativistic level though, this is my whole point.
Time for some math.
OK, what I’ve done here is kind of set myself up for what I’m trying to prove with the essential equations.
Back to my diagram in accordance with the math; remember that charge 2 is the moving charge. In the equation, the q’s represent the charge that is FEELING the effects electric/magnetic field, and I have defined that as charge 2. You might find that weird, why would charge 2 feel the electric field and magnetic field if it’s the one that’s moving and creating it. Well, I’m going off relativity here and I can see that relative to charge 2, the moving charge to a stationary observer, that charge is not moving and charge one is the only charge moving, in the opposite direction of course. That is why I declared the moving charge as in the negative direction. Therefore, mathematically we can say that . The magnetic field is the cross product of the velocity and magnetic field vector.
In a simplistic one loop diagram (more pertinent to charges on a quantum scale when traveling in trigonometric functions, but it still obviously suffices here), the magnetic field is perpendicular to the velocity and the force is orthogonal to that, and by using the right hand rule you can determine in which direction that vector is. The vector sum of the electric force and the magnetic force will be in the same plane.
Taking those equations I have above, I can do a lot of substituting and clearly come out with my current result. However, I need to explain the integral because it’s a bitch, and I will do that after I quickly substitute everything in. I was having a debate about this problem, this afternoon over lunch with some professors and currently my idea still holds out.
Substituting, E and B primarily, we now get:
That’s all beautiful except for that integral:
. My diagram sucks right here so I’m going to redraw it for any other normal Biot-Savart situation.
You have a wire, I was an idiot when I drew this and I’ll explain why. But this wire has electrons flowing through it rightward, another electron x distance away from the center of the wire will feel a force acting upon it relative to the sum of the magnetic field and electric field vectors. Yes, a magnetic field does affect an electron too. Electrons have a small magnetic dipole, or can act as a bar magnet, in a small way because they spin in a specific direction, and the direction of their polarity is relative to which way they spin; they spin with an integer number s which is relative to what orbital they are in an atom, but that’s pretty damn irrelevant to this, so I’ll stafoo. Back on topic, the reason this diagram is +1 on confusing is because I accidentally drew negative charges in the wire, which means whatever way your fingers curl using the right-hand rule, it’s the exact opposite because the charges are opposite, silly me.
Hear me out, ds, the part of the integral that is difficult to identify in my original situation with two lone electrons, can be seen in this wire diagram as an infinitesimally small portion of the wire. You infinitely add up tiny bits of the wire and calculate the forces for each little piece and add them up, you need to do this because the force is constantly INCREASING as you approach the mid point of the wire because the overall effect is related to 1/r^2, and since r is decreasing, the effect of that part of the equation is less significant. Going back to what I said in the beginning, electric fields only exist for where there is charge. Charge moves. Charge moves relativistically fast (speeds near the speed of light). This means that the vector ds, the infinitesimally small change in the wire, with respect to my first diagram of the two lone electrons, is ACTUALLY just the width of the electron. Because, get this, there will only be charge around the point that is the electron, therefore the diameter of the electron is exactly that part with respect to an instant in time. Therefore, we can now call the change in distance as just a constant as the width of an electron, a distance, if you will (therefore units are still matching up). R-hat is simply a unit vector with magnitude of 1 and we can call it +1 because at the end we’ll switch all the signs anyway. The sin theta of the cross product can be killed because our experiments will be done orthogonally. Therefore, our newest equation, and simplifying the other one, is (please note that w is equal to the width of the charge):
If w is a measurement of length of a charge (similar to distance), divided by time, and since v is the velocity of that same charge, we can combine like units into calling it:
Simplifying further, we get:
If you’re still reading and actually care, you might think this can be related to time dilation. Oh shit! It can, in an ugly way; I’m still working with the math making sure it’s true with Einstein’s equations.
All time dilation is, you need to understand conceptual relativity for this, but as an object is moving near the speed a light, the stationary observer sees its clock as traveling slower relative to its own, the ratio test regarding the difference between the rates of the two clocks can be known as the constant gamma.
All I need to do is connect them directly. I get:
I’ll keep it posted as I clean up the math. What this proves that on a relativistic scale, meaning almost all scales of normal charges regarding electrons and electric fields (the fields that surround those relativistic motions) magnetic fields arise. This means that magnetic fields are arbitrary and are just added to ease calculations but can be defined in terms entirely of electric fields and electric charges on a more fundamental scale. You’re a person without a life for reading all of this. I am for writing it, but I learned something.
You ask any physicist the definition of a magnetic field and he or she will either give you a textbook answer and or a mathematical answer. However, both are actually arbitrary. This is a little bit of a read but I urge you to read through all of it if you are going to make any comments or any attempt to criticize my thoughts.
I’ll repeat, I’m in the process of rendering the entirety of the subject of magnetism an arbitrary value. Does this mean people will stop teaching it? Absolutely not, because it’s so much more simplified (because of the fact it’s arbitrary) it will still go on being taught the same way; however, the goal here is to actually provide fundamental meaning of magnetism. What I mean by fundamental meaning to magnetism is prove that magnetism isn’t really anything but a simplified version of a very complex system of charges and electric fields.
My goal is to graphically, mathematically, and conceptually give reasoning for my theory in this blog; it may be a little disorganized but I have yet to actually organize it on paper yet so I have to start somewhere. In this part, part one of my explanations, a deep mathematical understanding of relativity isn’t necessary, just an abstract conceptual one.
I’m going to Photoshop an example diagram regarding my situation because that would look better than my MS Paint or blackboard drawings. It’s fairly abstract; however it’s also fundamental, which is what we’re aiming at. There are some obvious flaws but consider these findings in pre-beta version mathematically; conceptually it all makes sense, however much of it is a little counter-intuitive but will be explained later in some detail.
My labeling will seem counter-intuitive but when I get into the relativistic integration, it will make sense because gamma will need to be introduced sooner or later to make the correlation.
Picture two electrons, as I have, separated by distance ‘r’, where the second electron is traveling at velocity vector, in some arbitrary direction we will note as negative. Theta represents the angle between vector v and vector r. This isn’t too necessary but is required to be established here so it will make sense in the math later. Also, I’m restating the fact that I’m using relativity to define this therefore the fact that electrons have wave particle duality can be ignored because Euclidean geometry and Einsteinian mechanics work perfectly.
Think back a ways and remember what current is, aside from Ohm’s law, but on a more fundamental level. It is measured in amperes and is noted as the time derivative of charge (I’m going to do what everyone else does and label current, I, and charge, q): (try to picture this, it’s important because I make some nasty assumptions later). Well it’s also common sense that velocity is just the time derivative of a distance, r, . The definition of electric field, E, is the amount of force distributed for a certain charge and can be mathematically written as such: . Mathematically, it’s obvious that there is a direct correlation between electric field and current by the time derivative of charge. It can be written as follows: . We can rewrite the initial equation as therefore allowing us to establish an understanding of force which is how I will prove everything down the road.
Whenever there is the presence of a charge, there is an electric field that surrounds it and that electric field exerts a force on any surrounding charge proportional to the magnitude of something like 1/r^2. It’s 1/r^2 because we live in a three dimensional universe and everything expands outward from a center control with a radius. The only thing in three dimensions that corresponds to a radius that has equidistant unit points at every point in space, infinitesimally small units apart, is a sphere. Why not 1/r^3 if we’re in three dimensions? Think of the surface area of the sphere, it goes something like 4pi*r^2. When I get further into it, all the electrical constants have 4pi embedded into them somewhere, such as mu nought, coloumb’s constant, epsilon nought, etc. That’s where the r^2 comes from and it’s a very important concept to grasp. This is why magnetic field can’t be specifically described as “real” in the common sense because it goes by a proportional relationship like 1/r, not 1/r^2.
Taking this from Google’s definition feed, “magnetism is one of the phenomena by which materials exert an attractive or repulsive force on other materials”. Magnetism exists from the movement of electrical charge. Remember what I said earlier, the movement of electrical charge in a quantitative format is described as current. This needs to be defined on a relativistic level though, this is my whole point.
Time for some math.
OK, what I’ve done here is kind of set myself up for what I’m trying to prove with the essential equations.
Back to my diagram in accordance with the math; remember that charge 2 is the moving charge. In the equation, the q’s represent the charge that is FEELING the effects electric/magnetic field, and I have defined that as charge 2. You might find that weird, why would charge 2 feel the electric field and magnetic field if it’s the one that’s moving and creating it. Well, I’m going off relativity here and I can see that relative to charge 2, the moving charge to a stationary observer, that charge is not moving and charge one is the only charge moving, in the opposite direction of course. That is why I declared the moving charge as in the negative direction. Therefore, mathematically we can say that . The magnetic field is the cross product of the velocity and magnetic field vector.
In a simplistic one loop diagram (more pertinent to charges on a quantum scale when traveling in trigonometric functions, but it still obviously suffices here), the magnetic field is perpendicular to the velocity and the force is orthogonal to that, and by using the right hand rule you can determine in which direction that vector is. The vector sum of the electric force and the magnetic force will be in the same plane.
Taking those equations I have above, I can do a lot of substituting and clearly come out with my current result. However, I need to explain the integral because it’s a bitch, and I will do that after I quickly substitute everything in. I was having a debate about this problem, this afternoon over lunch with some professors and currently my idea still holds out.
Substituting, E and B primarily, we now get:
That’s all beautiful except for that integral:
. My diagram sucks right here so I’m going to redraw it for any other normal Biot-Savart situation.
You have a wire, I was an idiot when I drew this and I’ll explain why. But this wire has electrons flowing through it rightward, another electron x distance away from the center of the wire will feel a force acting upon it relative to the sum of the magnetic field and electric field vectors. Yes, a magnetic field does affect an electron too. Electrons have a small magnetic dipole, or can act as a bar magnet, in a small way because they spin in a specific direction, and the direction of their polarity is relative to which way they spin; they spin with an integer number s which is relative to what orbital they are in an atom, but that’s pretty damn irrelevant to this, so I’ll stafoo. Back on topic, the reason this diagram is +1 on confusing is because I accidentally drew negative charges in the wire, which means whatever way your fingers curl using the right-hand rule, it’s the exact opposite because the charges are opposite, silly me.
Hear me out, ds, the part of the integral that is difficult to identify in my original situation with two lone electrons, can be seen in this wire diagram as an infinitesimally small portion of the wire. You infinitely add up tiny bits of the wire and calculate the forces for each little piece and add them up, you need to do this because the force is constantly INCREASING as you approach the mid point of the wire because the overall effect is related to 1/r^2, and since r is decreasing, the effect of that part of the equation is less significant. Going back to what I said in the beginning, electric fields only exist for where there is charge. Charge moves. Charge moves relativistically fast (speeds near the speed of light). This means that the vector ds, the infinitesimally small change in the wire, with respect to my first diagram of the two lone electrons, is ACTUALLY just the width of the electron. Because, get this, there will only be charge around the point that is the electron, therefore the diameter of the electron is exactly that part with respect to an instant in time. Therefore, we can now call the change in distance as just a constant as the width of an electron, a distance, if you will (therefore units are still matching up). R-hat is simply a unit vector with magnitude of 1 and we can call it +1 because at the end we’ll switch all the signs anyway. The sin theta of the cross product can be killed because our experiments will be done orthogonally. Therefore, our newest equation, and simplifying the other one, is (please note that w is equal to the width of the charge):
If w is a measurement of length of a charge (similar to distance), divided by time, and since v is the velocity of that same charge, we can combine like units into calling it:
Simplifying further, we get:
If you’re still reading and actually care, you might think this can be related to time dilation. Oh shit! It can, in an ugly way; I’m still working with the math making sure it’s true with Einstein’s equations.
All time dilation is, you need to understand conceptual relativity for this, but as an object is moving near the speed a light, the stationary observer sees its clock as traveling slower relative to its own, the ratio test regarding the difference between the rates of the two clocks can be known as the constant gamma.
All I need to do is connect them directly. I get:
I’ll keep it posted as I clean up the math. What this proves that on a relativistic scale, meaning almost all scales of normal charges regarding electrons and electric fields (the fields that surround those relativistic motions) magnetic fields arise. This means that magnetic fields are arbitrary and are just added to ease calculations but can be defined in terms entirely of electric fields and electric charges on a more fundamental scale. You’re a person without a life for reading all of this. I am for writing it, but I learned something.
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