Submitted:
14 February 2025
Posted:
18 February 2025
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Abstract
The difference in velocity between the electrons and protons in an electrical conductor is said to produce a magnetic force due to Special Relativity effects. According to established theory, the magnetic force observed from a stationary reference frame is that of the electrostatic force viewed from a moving reference frame in consideration of Special Relativity effects. The traditional method of calculating the electrostatic force from a moving reference frame and considering Special Relativity effects is laborious. The approach includes calculation of the force between two parallel electrical conductors. This paper presents a more direct approach for calculating the electrostatic field that involves only a single electrical conductor. The new approach also provides a clearer understanding of the underlying physics.
Keywords:
Magnetic fields
; Special Relativity
; Linear charge density
Introduction
The difference in velocity between the electrons and protons in an electrical conductor is said to produce a magnetic force due to Special Relativity effects. According to established theory, the magnetic force observed from a stationary reference frame is that of the electrostatic force viewed from a moving reference frame in consideration of Special Relativity effects.
The traditional mathematical approach for calculation of the Special Relativity effects includes calculation of the attraction between two parallel wires with electrical current flowing equally through them in the same direction [1,2]. The calculation is performed from a moving reference frame for consideration of the Special Relativity effects.
This paper presents a more direct calculation of the electrostatic field arising from Special Relativity effects. With the approach presented in this paper, only a single wire is needed for the calculation. The present approach is less laborious and provides a clearer understanding of the underlying physics.
This paper begins with review of the magnetic field equation for electrical conductors, and review of electrostatic charges in copper wire conductors.
Magnetic field around an electrical conductor
According to Ampere’s Law, the magnetic field density (B) around an electrical conductor is equal to,
where μo is the permeability of free space, I is the electrical current, and R is the radial distance from the electrical conductor.
B = μoI/2πR
Copper wire electrical conductors
Copper atoms have an equal number of electrons and protons. When viewed from a stationary reference frame, the negative static electric charge from the electrons is exactly canceled out by the positive static electric charge of the protons, leaving the copper atoms with a net static electric charge of zero.
The protons are secured in place in the atom’s nucleus, and not free to leave the copper atom. By contrast, one of the outer valance electrons is free to leave the atom, provided that a second electron is added to the atom to take its place. This free motion of the outer electrons enables the flow of electron current through copper wire. So long as the departing electron is replaced by an arriving electron, the static electric charge of the copper remains equal to zero. Electrical current flowing through copper wire does not produce a static electric charge when viewed from a stationary reference frame.
Static electric linear charge density
The negative flux from the electrons is exactly cancelled out by the positive flux from the protons. Let’s calculate the flux density from the electrons alone in the copper wire to solve for how large these balanced push-and-pull forces are.
The static electric charge from the free electrons in a length of copper wire is equal to,
where qe is the static electric charge of an individual electron, nc is the number of free electrons per cubic meter of copper [3], Ab is the bare copper wire cross sectional area in meters squared, and LS is the length of the wire in meters, where the subscript “s” denotes a stationary reference frame.
Qe = qencAbLS Units: (C/e)(e/m3) m2 m = C
qe = -1.602,177 x 10-19 C Units: C
nc = 8.49 x 1028 Copper Units: e/m3
Static electric flux density E is equal to the charge Qe divided by the surface area that the charge radiates through and the permittivity constant εo. The area A of a cylinder of radius R surrounding the wire is equal to,
A = 2πRLS Units: m2
The static electric flux density from the electrons (ESe) at distance R from the copper wire is then,
ESe = Qe/Aεo Units: C/m2
= qencAbLS/2πRLSεo
ESe = qencAb/2πRεo
The subscript “Se” for flux density ESe denotes a stationary reference frame for the electrons.
Let’s define a new term, linear charge density, denoted “λ”, where the linear charge density for the free electrons alone (λSe) is equal to,
λSe = qencAb Units: (C/e)(e/m3)(m2) = C/m
Equation 7 can then be simplified to,
ESe = λSe/2πRεo
The linear charge density for the associated protons (λSp) is equal in magnitude and opposite in sign to that of the electrons (subscript “s” denotes a stationary reference frame, and subscript “p” denotes protons). When the proton linear charge density λSp is added to the electron linear charge density λSe the net linear charge density (λSN) equals zero:
λSN = λSp + λSe = 0
And,
λSp = -λSe
The net static electric linear charge density observed from the stationary reference frame (ESN) is then,
ESN = λSN/2πRεo = 0
As Equation 12 indicates, electrical current flowing through copper wire does not produce a static electric charge when viewed from a stationary reference frame.
Electrical current
The electrical current flowing through copper can be solved for in terms of electron charge qe, free electron density in copper nc, wire cross sectional area Ab and drift velocity vd, where,
I = qencAbvd Units: (C/e)(e/m3)(m2)(m/s) = C/s
Drift velocity vd is in units of m/s. Units of meters squared are used for area Ab, and the number of free electrons in a cubic meter of copper nc is provided by Equation 4. Substituting in Equation 8 for linear charge density λSe we have,
I = λSevd = qencAbvd Units: C/s
Time dilation according to Special Relativity
Time dilation is the stretching of time for objects in motion. The faster an object moves, the slower the object ages.
Let’s describe time dilation by way of an example, where Tom conducts a measurement of time on a moving train. The time measured by Tom while he is traveling on the train is denoted “tT”. Sally conducts the same measurement of time but she is standing at the train station. The time measured by Sally from the stationary train station is denoted “tS”.
Tom’s experiment includes a light source, a reference table, a mirror and a clock. The mirror is mounted on the ceiling above the reference table. With the clock, Tom measures the time tT for a pulse of light to travel vertically from the reference table, bounce off the mirror, and return vertically to the reference table.
The distance between the reference table and ceiling is denoted D, and the speed of the light is denoted c. Knowing that rate times time equals distance, the up and down distance is equal to,
ctT = 2D
And dividing Equation 15 by the speed of light c, Tom’s measured time tT is equal to,
tT = 2D/c Time measured by Tom on the train
Sally measures the elapsed time as well, but she uses two synchronized clocks located on the stationary train station platform. One clock is located adjacent to the train where the light starts, and the other clock is located adjacent to the train where the light ends. From Sally’s perspective the light travels vertically distance D twice (up then down), plus the horizontal distance the train traveled down the train track (h). The horizontal distance traveled down the train track h is equal to the velocity of the train (vS) observed by Sally, times the time measured by Sally tS, where,
h = vStS
From Sally’s perspective the light travels diagonally up then down due to the vertical motion between the reference table and ceiling of the train, and the horizontal motion due to the train moving down the track. Let’s denote each of the diagonal lengths “L”. The diagonal distance L as seen by Sally is found using the Pythagorean Theorem, where,
L = [(h/2)2 + D2]0.5 ≡ [(vStS/2)2 + D2]0.5
From Equation 15, distance D equals,
ctT/2 = D
Substituting this value for distance D into Equation 18 we have,
L = [(vStS/2)2 + (ctT/2)2]0.5
Alternatively, Sally could use her measurement of time tS to calculate the distance traveled by the light using the simple relationship of rate times time equals distance. For her measured time of tS and the speed of light c we have the total distance traveled 2L equal to,
2L = ctS
Or rearranging terms to solve for the time measured by Sally from her stationary train platform:
tS = 2L/c Time measured by Sally from the train station
Substituting distance L from Equation 19 into Equation 21 to solve for the time observed by Sally tS from the stationary train station platform we have,
tS = 2[(vStS/2)2 + (ctT/2)2]0.5/c
Simplifying Equation 22 we have,
ctS/2 = [(vStS/2)2 + (ctT/2)2]0.5
(ctS/2)2 = (vStS/2)2 + (ctT/2)2
(ctS/2)2 - (vStS/2)2 = (ctT/2)2
(ctS)2 - (vStS)2 = (ctT)2
c2tS2 - vS2tS2 = c2tT2
tS2 - (vS2/c2)tS2 = tT2
tS2(1 - (vS2/c2)) = tT2
tS2 = tT2/(1 - (vS2/c2))
(ctS/2)2 - (vStS/2)2 = (ctT/2)2
(ctS)2 - (vStS)2 = (ctT)2
c2tS2 - vS2tS2 = c2tT2
tS2 - (vS2/c2)tS2 = tT2
tS2(1 - (vS2/c2)) = tT2
tS2 = tT2/(1 - (vS2/c2))
tS = tT/[1 - (vS/c)2]0.5 Time dilation
To reduce the number of terms that need to be typed in Equation 23, Lorentz defined the Lorentz Factor (γ):
γ = 1/[1 - (vS/c)2]0.5 Lorentz factor
Substituting Equation 24 into Equation 23, the time observed by Sally from the stationary train station platform tS is then equal to,
tS = tTγ Time dilation
γ is non-dimensional, as the units for vS/c cancel out. γ can be thought of as a scalar magnitude. γ is also always equal to or greater than 1, and when the velocity vs equals zero the Lorentz Factor equals 1:
γ ≥ 1
γ = 1 When vs = 0
Length Contraction
A related effect of time dilation is length contraction. We will use a moving train and Sally and Tom’s reference frames again to describe length contraction. This time, let’s assume Sally has a ruler of length LS extending lengthwise in the direction of the train track.
Tom passes by Sally at velocity vS. In Sally’s reference frame the time for Tom to travel the length of her ruler LS is,
tS = LS/vS Time measured by Sally
Tom’s clock runs slower due to time dilation. Therefore, the time measured by Tom equals,
tT = tS/γ
Looking out the window, Tom measures the time to pass the ruler, where,
tT = LT/vS Time measured by Sally
Replacing Tom’s time tT with Equation 28 we have,
tS/γ = LT/vS
tS = γLT/vS
Replacing Sally’s time tS with Equation 27 we have,
LS/vS = γLT/vS
LS = γLT
Because the Lorentz Factor γ is greater than 1, the length measured by Tom is shorter than the length measured by Sally.
Sally’s view of the wire
Let’s now assume instead of a train Sally is observing a copper wire having an electron current. When Sally views the copper wire she observes the electron current flowing to the right with drift velocity vd. When Sally measures the static electric field next to her she records a value of zero, because there is an equal number of electrons and protons in a given length of wire.
Tom’s view of the wire
Next, let’s imagine Tom on a train moving at drift velocity vd to the right. From his perspective the electrons are stationary and the protons are moving to the left at velocity vd.
Because Tom is moving, his clock is also ticking slower. His second takes longer than Sally’s second. The length of Tom’s second is found by time dilation Equation 25. Because of his longer second, Tom observes more protons passing by than a person would without time dilation. Tom observes a proton linear charge density (λTp) equal to,
λTp = λSpγ
Where the Lorentz factor γ is calculated using Tom’s drift velocity vd. Subscript “T” denotes a traveling reference frame.
Tom also experiences length contraction due to his velocity. Accordingly, Tom’s ruler becomes shorter than Sally’s ruler. When Tom measures the spacing between the electrons next to him, he observes a larger spacing than Sally does, because his ruler is shorter. The larger spacing between the electrons means that Tom observes a smaller linear electron charge density (λTe) than what Sally observes. Tom observes an electron linear charge density (λTe) equal to,
λTe = λSe/γ
Where the Lorentz factor γ is calculated using Tom’s drift velocity vd. The net linear charge density (λTN) observed by Tom is the sum of the positive proton linear charge density λTp and the negative electron linear charge density λTe, where,
λTN = λTp + λTe
Replacing linear charge densities λTp and λTe with Equations 32 and 33 we have,
λTN = λSpγ + λSe/γ
Replacing λSp with Equation 11 we have,
λTN = -λSeγ + λSe/γ
λTN = λSe(-γ + 1/γ)
The net electrostatic field density observed by Tom (ETN) is then equal to,
ETN = λTN/2πRεo
And recalling from Equation 11, we have the net electrostatic field density observed by Sally (ESN),
ESN = λSN/2πRεo = 0
Equation 37 provides the electrostatic field density observed from a moving reference frame with consideration of Special Relativity effects, that is perceived as a magnetic field density when observed from a stationary reference frame.
Summary
The difference in velocity between the electrons and protons in an electrical conductor is said to produce a magnetic force due to Special Relativity effects. According to established theory, the magnetic force observed from a stationary reference frame is that of the electrostatic force viewed from a moving reference frame in consideration of Special Relativity effects. The traditional method of calculating the electrostatic force from a moving reference frame and considering Special Relativity effects is laborious. This paper presents a more direct approach for calculating the electrostatic field that involves the following four short equations:
λSe = qencAb Electron linear charge density
γ = 1/[1 - (vS/c)2]0.5 Lorentz factor
λTN = λSe(-γ + 1/γ) Net linear charge density
ETN = λTN/2πRεo Net electrostatic field density
The net electrostatic field density ETN due to Special Relativity effects is found to be due to the electron drift velocity, where electron drift velocity vS in Equation 24 is measured from Sally’s stationary reference frame.
References
- Purcell, Edward and Morin, D.: Electricity and Magnetism, Third Edition, Cambridge University Press 2013, page 259.
- Griffiths, David: Introduction to Electrodynamics, Fourth Edition, Cambridge University Press, 2017, pages 242, 550.
- Walker, Jearl: Fundamental of Physics, 9th Edition, Wiley pub 2011, page 688.
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