Significant Relativistic-Effect of Insignificantly Low-Velocity !

R.C. Gupta Visiting Professor, Institute of Engineering & Technology (I.E.T.), Lucknow, (iitk Alumni), India ([email protected]) & Ruchi Gupta Innovator, Strategist, Product Management, (Stanford Alumni), San Francisco/Bay Area,CA, USA ([email protected])

Abstract

Though it is not obvious, but the truth is that operation of the electric motor (or fan) & many household appliances is based on relativistic effect of electrons moving with insignificantly low drift-velocity. If velocity is insignificantly small, such as electron’s drift velocity in current flow, of the order of snail’s speed (< 1 mm/s) or v/c = 0.3 x 10-10 ; its corresponding relativistic effect factor ½ v2/c2 will be too small of the order of 10-21 certainly seems a negligible factor. In the present paper it is shown that, in totality, the relativistic effect of such insignificantly low-velocity could be significant, noticeable & observable too. It is asserted in the paper that such significant, noticeable & observable effect, is in-fact the well known to us (lab observer) as ‘magnetism’! Although the full story of how special-relativity links electricity and magnetism is mathematically quite complex, some aspects of it are easy to appreciate if one tries to answer “Why two parallel wires carrying currents in same direction attract each other ?” The conventional answer from lab- observer perspective is that the current in wire-1 produces a magnetic-field B according to Biot-Savart law around the wire in the circumferential-direction as per right-hand screw, this magnetic-field B then acts on wire-2 which results-in the attractive force towards wire-1 according to Fleming’s Left-hand rule. Whereas from the perspective of the moving electrons in wire-1, the positive-charges in the wire-2 appear to be moving away and thus the little length-contraction as-if produces more positive-charges therein and thus net extra attractive force. This apparent magnetism, in reality is due to Coulomb’s electrostatic attraction between a pair of electron (in wire-1) & extra-positive-charge (in wire-2), though much less by the factor 10-21 ; but since there may be 1020 or more electrons & positive-charges per centimeter in such a wire, so the total collective net effect would be significant & noticeable.

1. Introduction

‘The most important revolution in the history of civilization is ‘industrial revolution’ due to appreciation of steam-power to make steam-engines & steam-turbines. The next important inventions are the electric-generator & electric-motor(fan) which gave a boost to the industrial-revolution’. This is what an engineer might be thinking, looking from the side-window seeing the various machines running with its electric-motors while sitting in his room wherein a fan is running to provide some cooling-comfort in hot- summer. But though he & we know that the working of electric generator, motor, fan & many household appliances is due to electromagnetic-induction; probably most people don’t know that the ‘magnetism’ to the lab- observer/engineer (we/he) in-fact is special-relativistic effect (length contraction) due to electrons moving (or relatively positive-charges moving away in opposite-direction) with insignificantly low drift- velocity, but in the electron’s frame of reference this (magnetism) is due to collective Coulomb’s electrostatic attraction between many pairs of electrons in one-wire & extra-positive-charges due to length- contraction (shortening of spacing between the successive positive-charges) in the other-wire. Similarly, to the 1 stationary positive-charge in one-wire the electron are moving in the other-wire, hence length-contraction of spacing between successive electrons thus the extra-electrons results in attraction between the two wires. In- fact, it is the Special-Relativity [1,2] which provides a ‘bridge’ between Electricity and Magnetism [3-5].

2. Relativistic Effects:

2 2 1/2 Special-relativity results such as the formulae for: length-contraction L = L0.(1 – v /c ) , 2 2 1/2 2 2 1/2 time-dilation t = t0 /(1 – v /c ) and mass-increase m = m0 /(1 – v /c ) depend on the relative velocity v between the two frames of reference. The differences L0 – L , t – t0 & m –m0 due to the relativistic formulae 2 2 2 2 2 2 are nearly ½ v /c L0, v /c t0 & ½ v /c m0 respectively. Electron’s orbital speed v in atom is about 1% of light-speed c, hence the relativistic-effect which could be there by a factor ½ v2/c2 will be very low by a factor ½ (0.01)2 = 0.00005 which is very small & is almost unnoticeable. What if velocity is insignificantly small, such as electron’s drift-velocity [3,4] in current flow, of the order of snail’s speed (< 1 mm/s) or v/c = 0.3 x 10-11 ; its corresponding relativistic effect factor ½ v2/c2 will be too small of the order of 10-23 certainly seems a negligible factor. But in the present paper it is shown that, in totality, the relativistic effect of such insignificantly low-velocity could be significant, noticeable & observable too. It is asserted in the paper that such significant, noticeable & observable effect, is in-fact the well known to us (lab observer) as ‘magnetism’! This apparent magnetism, in reality is due to Coulomb’s electrostatic attraction between a pair of electron in one-wire & extra-positive-charge in the other-wire (similarly, attraction between the positive-charge in one wire & extra electrons in the other wire), but much less by length-contraction-factor 10-23 ; but since there may be 1020 or more electrons & positive-charges per centimeter [4] in such wires, so the total collective effect (10-3) may be significant & noticeable since electrostatic-force is much stronger [4] than gravitational-force by a factor 1039. Though it is not obvious, but the Truth is that ‘operation of a electric motor (or fan) is based on relativistic effect of electrons moving with insignificantly low drift-velocity’.

3. Relativity is the bridge between Electricity and Magnetism

Although the full story of how special-relativity links electricity and magnetism is mathematically quite complex [3], some aspects/examples of it are easy to appreciate, and that is what is done in the literature/books [3-5] and also in the present paper but with more generality & clarity. An example is the attractive force between two parallel conductors (wires 1 & 2) carrying current in same directions. How we tackle the problem, for this there are two perspectives viz., (1) Electric-charge’s perspective and (2) Lab- observer perspective. An observer (as if) sitting on an electron moving with the electrons drift velocity v unaware of magnetism, finds electrostatic Coulomb’s attraction due to length contraction (hence more positive-charges in the other wire). Similarly, stationary positive-charge in one-wire finds more moving- electrons in the other-wire, hence attraction. On the other hand, the lab-observers (we, you) think that the wire-1 produces a magnetic field (B) which acts on the moving electrons in the wire-2 and thus produces a Lorentz force F = Q.v x B towards wire-1 as per Fleming’s left hand rule (also applicable to the working of electro-motor/fan). The final result in both frame of references is same, a force per unit length between two parallel wires carrying currents I1 and I2 in same direction comes out as F/L = - µ/(4π) 2 I1.I2/d , the negative sign implies attractive force. The detailed derivation of the formula from both the perspectives are given in the next section 4(a) & 4(b). Thus, it is often said that ‘Special-Relativity’ is the bridge which connects ‘Electricity’ and ‘Magnetism’.

4(a). Force between the two parallel current carrying conductors (wires) W1 & W2 from the perspective (or frame of reference) of charges in the wire W1

Consider the two current carrying long wires W-1 & W-2 at a distance d with currents (right to left) I1 & I2. In wire W-1, the positive-charges (P1) are non-moving but the negative-charge electrons (E1) are moving (left to right) with drift velocity v1. Similarly, in wire-2, positive-charges (P2) are non-moving but the electrons (E2) are moving (left to right) with drift velocity v2 (Fig.1.b). The perspective from wire-1 (i.e., from the perspective of E1 & P1) is considered here, as follows; wherein the relativistic length- 2 contraction factor leading to extra Coulomb’s attraction are indicated, -ive sign for attraction and +ive sign for repulsion (Fig.1 c & d). 2 2 E1 seeing E2 + ½ (v2- v1) /c repulsion 2 2 E1 seeing P2 - ½ v1 /c attraction P1 seeing P2 0 Nil 2 2 P1 seeing E2 - ½ v2 /c Attraction 2 2 2 2 2 Total = ½ [(v2- v1) – v1 – v2 ] / c = - v1.v2 / c Attraction

Net extra attractive normal (perpendicular to wires) Force dF between the two wire elements dL1 & dL2 (the line joining these elements makes an angle θ with the wires) is given as follows, to be integrated to give F. It is interesting to note that starting from Coulomb’s formula, the first-integral is the formula for Biot-Savart Law for magnetic-field B created by the current I1 wire-1, whereas incorporating the second-integral gives Ampere-force & Lorentz-force

2 2 2 dF = - (1/4πε).dq1.dq2/r . v1.v2 / c . sin θ , extra Coulomb’s force (attraction) where c = 1/(ε.µ) 2 F = - ∫∫ (µ/4π). 1/r .(dq1. dL1/dt).(dq2. dL2/dt). sinθ ; using v1= dx1/dt = dL1/dt, v2= dx2/dt = dL2/dt 2 = - ∫ (µ/4π). 1/r .(dq1/dt. dL1. sin θ) x ∫ (dq2/dt.dL2) , rearrangement of terms, x for cross product 2 = - ∫ (µ/4π). {1/r . I1 . dL1. sin θ} x ∫ I2.dL2 , the ‘first integral’ is Biot-Savart Law for B = - (µ/4π). {1/d . 2I1} x ∫ I2.dL2 , after the integration & simplification for ‘long’ wire-1 = - (µ/4π). 2I1/d x I2. L2 , the Ampere’s force law between two current-carrying conductors = - B x.I2.L2 , like the Lorentz formula, the ‘first integral’ is for magnetic field B = (µ/4π). 2I1 / d

= - B x ∫ dq2 / dt . dL2 = - B x ∫ dq2 . dx2/dt = -B x Q.v2 = Q. v x B Lorentz magnetic force formula

4(b). Force between the two parallel current carrying conductors (wires) W1 and W2 from the perspective (or frame of reference) of Lab-observer

From wire-1 perspective, only the extra Coulomb’s forces due to relativistic length-contractions have been considered. Magnetic force not included therein, because it is this extra Coulomb’s field of wire-1 which appear acting on wire-2 in the form of the magnetic field in the lab-observer’s perspective to give the Lorentz magnetic force.

But from lab-observer’s perspective, there would / could be two forces (Lorentz and Coulomb’s) to be considered as follows:

(i) The current in wire-1 produces a magnetic field B = (µ/4π). 2I1 / d which acts on the moving charge at the station-2 producing a magnetic force F = Q. v x B (use the Lorentz formula) as shown in Fig.1.e . (ii) The lab-observer will, however, also notice, in his-frame, see Fig.1.b, new length- contractions (which may or may-not add up to zero) for the extra Coulomb’s contribution, if there.

(i) Lorentz Force FL due to Magnetic-field B from the lab-observer’s perspective

FL = - B x ∫ dq2.v2 = - B. ∫ I2. dL2 = - B x I2. L2 , where B = (µ/4π). 2I1 / d = - (µ/4π). 2I1/d x I2. L2 Ampere force law (attraction)

3 (ii) Coulomb’s Force FC due to Relativistic length-contraction from the lab-observer’s perspective

The new length contraction factors from the lab-observer (L) perspective to see the electrons & positive-charges of ‘both’ the wires (1 & 2) are tabulated below which can contribute for extra Coulomb’s attraction or repulsion. In this case, it can be seen that the total of the relativistically induced

interaction between electrons (E1, E2) & positive-charges (P1,P2) for Coulomb’s force FC finally adds

up to zero. The Grand- Total thus is equal to FL, which is the Ampere’s force, same as derived in 4(a).

2 2 L seeing E1 yields length-contraction for electrons in wire-1 as ½ v1 /c (Fig.1.b)

2 2 and L seeing E2 yields length-contraction for electrons in wire-2 as ½ v2 /c (Fig.1.b)

Hence , the relativistic force ‘factors’ to contribute to extra Coulomb’s attraction or repulsion are:

2 2 2 2 L finds relativistic force factor between E1 & E2 as ½ v1 /c + ½ v2 /c (repulsion)

2 2 L finds relativistic force factor between P1 & E2 as - ½ v2 /c (attraction)

L finds relativistic force factor between P2 & P1 as 0 (nil)

2 2 L finds relativistic force factor between E2 & P1 as - ½ v1 /c (attraction)

total FC = 0 incidentally (nil)

Grand Total of the two forces of (i) and (ii) = FL + FC = - (µ/4π). 2I1.I2. L2 / d (attraction) Ampere’s law

(‘same result’ as found in earlier in section 4(a) )

5. Discussion

What is the ‘actual’ average drift-velocity of the electrons in a typical current-flow ? It appears in the first instance that it must be very high of the order of speed of light because in the electric-circuit the electric-bulb almost immediately gives light as soon as the switch is made ‘on’. In-fact it is the electric-field that travels very fast almost at speed of light that lightens the bulb as soon as the switch is made ‘on’. In the books / literature it has been estimated that the actual average drift-velocity of electrons is surprisingly very small of the order of snail’s speed (v < 1 mm/s). In the present paper it has been shown though electron’s drift-velocity is insignificantly small, it relativistic-effect is significant and observable too; it is the ‘collective Coulomb’s electrostatic force’ in electron’s frame of reference which appears as observable ‘magnetism’ to the lab-observer.

To remove some apparent skepticism in the word “seeing” used in sections 4 (a) & (b), the authors urge to say as follows. “Seeing” does not simply mean ‘seeing’ by living (human) or by non-living (camera), the observer could be anything. It has wide ranging meaning & implications encompassing quantum- mechanics & philosophy, wherein importance of ‘seeing’ has been debated enough at length. ‘Seeing’ may mean ‘to love (attraction) & hate (repulsion)’ or ‘to be loved & hated’. It also means ‘observation’ or ‘measurement’. Here, ‘seeing’ implies in a sense ‘to observe or to feel’ so as ‘to influence the others or to be influenced by the others’. ‘Seeing’ is not only done by eyes, it can be ‘listened’ (friends listen even unsaid words !) or its ‘influence’ can be ‘felt’ through ‘contact’ or through ‘action at a distance’. ‘Seeing’ is also neurologically linked to ‘thinking’ or ‘imagining’ (the well known medical phrase: ‘the eyes see what the mind knows’). That’s why the ‘thought experiments’ are permitted in science, especially 4 in physics. To ‘imagine’ as if ‘seeing or observing’, is a common & well-accepted practice in ‘special- relativity’. There is no harm in ‘seeing’ or ‘to do whatsoever’, if it is intended to reveal Truth; and ‘that is what’ has been done in the standard text-books and here too. ‘All avenues for Truth must be kept open, at least it should not be obstructed’ !

In fact, there are basically two perspectives. The first perspective 4(a) is from wire-1- frame (i.e., the charges therein, specially) or equivalently from wire-2-frame, and the second perspective 4(b) is from the lab-observer-frame. In the first perspectives (frames), only the relativistic- effects (length-contractions) play towards the Coulomb’s attraction/repulsion role; which appears to the lab-observer as magnetic-field (B) generated by the wire-1 acting on the wire-2 to produce the magnetic force (using the famous Lorentz formula F = Q. v x B ) , the lab-observer in his-frame will also notice new length-contractions (but total may or may-not be zero) induced Coulomb’s contribution in addition to the magnetic-force. Interestingly, from both the perspectives, the net results should be the same (this works as a short of reconfirmation of the results). As an illustration, clarification & understanding the derivation of Ampere’s law for the force between two current-carrying conductors (wires) from both the perspectives are given in sections 4(a) & 4(b), the results of both the perspectives are same.

6. Conclusions

It is true that usually relativistic-effect is important & significant only when velocity is very very high such as in the particle-accelerators. Low-velocity relativistic-effect being too small is negligible, usually; but not always ! Example of ‘significant relativistic-effect of insignificantly low-velocity’ is what we call as ‘magnetism’ which is important ingredient for running of electric-machines such as motors &fans. It is illustrated ‘how two parallel wires carrying current in same direction are attracted to each other’. The lab-observer (section 4(b)) says that magnetic-field B generated by wire-1 according to Biot-Savart formula, acts on the wire-2 thus creates an attractive-force as per Fleming’s Left-hand rule between the two wires. To the moving electrons in wire-1 (section 4(a)) perspective (frame) the positive-charges in the other-wire appear to be relatively moving-away; thus the length-contraction implies shrinking of the distance between two successive positive-charges or in other words as-if extra positive-charges appear in wire-2; therefore small attraction through Coulomb’s formula is created, but since there are large number of such electrons & positive-charges are therein, the net attraction is significant. Similarly, to the positive charge (in wire-1) perspective too, as shown in section 4(a), there will be attraction between the two wires. In-fact, ‘magnetism’ is another facet of ‘electricity’ via ‘special-relativity’.

Acknowledgement

The authors are thankful to Dr. M.S. Kalara, Emeritus-Professor, I.I.T. Kanpur, Dr. A. Pradhan, Professor, GLA University, Mathura, Dr. B. Das, Professor, University of Lucknow and to his colleagues Dr. Rajiv Kumar, Prof. Arun Mittal , Dr. H.K. Paliwal, Dr. Shailendra Sinha, Prof. K.K . Srivastava and Prof. H.N. Gupta of I.E.T. Lucknow for valuable discussions & opinions. Thanks are also due to Sanjay Gupta, Shefali Gupta, Sushant Gupta & Shubham Gupta for assistance.

References

1. Albert Einstein, ‘Relativity: The Special and General Theory’, Wings Books, Random House, New York, 1961. 2. Albert Einstein, ‘The Meaning of Relativity’, 5th Ed., MJF Books, New York, 1984. 3. David J. Griffiths, ‘Introduction to Electrodynamics’, PHI learning, 2013, pp.210-226. 4. Arthur Beiser, ‘Concept of Modern Physics’, Mc Graw Hill, 1995, pp.19-21. 5. http://physics.weber.edu/schroeder/mrr/MRRtalk .

5 Figure 1 Force between two current carrying conductors from two different perspectives (frame of references) as explained in sections 4(a) and 4(b). For visual clarity, length-contraction(s) shown here are exaggerated.

Fig.1a Shows two wires 1 & 2 without current wherein equal number of negative-charges (electrons) & positive-charges are shown equi-spaced. Fig.1b Shows two wires 1 & 2 carrying current I1& I2 as seen by a lab-observer; wherein the electrons are moving with velocity v1& v2 , now due to length-contraction the spacing between moving electrons shortens and hence more number of electrons appear therein. Fig.1c From the moving electron perspective in wire-1, positive-charges in wire-2 seem to be moving in opposite direction with velocity v1 and thus due to length-contraction the spacing between two positive-charges shortens thereby more positive-charges appear in wire-2 which results-in the attraction between the two wire. Whereas the electrons in wire-2 seem to move right-to-left with velocity v1 – v2 thus little-bit of length-contraction. Fig.1d From the stationary positive-charge perspective in wire-1, electrons in wire-2 are moving left-to- right with velocity v2 and thus due to length-contraction the spacing between two electrons shortens thereby more electrons appear in wire-2, which results-in the attraction between the two wires. Fig.1e To the Lab-observer; current I1 in wire-1 causes a circumferential magnetic-field B as per right- hand-screw rule which acts on current I2 in wire-2 which results-in the attraction of wire-2 toward wire-1 as per Fleming’s Left-hand-rule.