A proposed test of special relativistic mechanics at low speed 1 2 4 5 6 7 8 9 10 1 2 13 14 15 16 17 18 19 20 21 22 2 3 3132 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 5757 5[.]
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8 Electrical and Computer Systems Engineering, School of Engineering, Monash University, 47500 Bandar Sunway, Malaysia
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1 2 a r t i c l e i n f o
13 Article history:
14 Received 8 November 2016
15 Accepted 24 December 2016
16 Available online xxxx
17 Keywords:
18 Charged particle
19 Uniform magnetic field
20 Special-relativistic mechanics
21 Newtonian approximation
22
2 3
a b s t r a c t
24
We show that the difference between the Newtonian and special-relativistic predictions for the angular
25 position increases linearly with time for a charged particle moving at low speed in a circular path in a
26 constant uniform magnetic field Numerical results suggest that it is possible to test the two different
pre-27 dictions experimentally
28
Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://
29
creativecommons.org/licenses/by-nc-nd/4.0/)
30
31 Introduction
33 Recently, it was shown numerically for a dissipative bouncing
34 ball system that, although the speed of the ball is low and the
35 gravitational field is weak, the Newtonian approximation to the
36 chaotic general-relativistic trajectory breaks down rapidly [1]
37 The different Newtonian and general-relativistic chaotic
trajecto-38 ries could be tested in the laboratory but the parameters and initial
39 conditions of the system must be known to very high accuracies so
40 that sufficiently accurate trajectories can be calculated for
compar-41 ison with experiment[2] Similarly, for low-speed non-dissipative
42 systems where gravity does not play a dynamical role, it has been
43 shown that the special-relativistic trajectory is not always
well-44 approximated by the Newtonian trajectory, regardless of whether
45 the trajectories are chaotic or non-chaotic [3,4] However, these
46 systems are model systems[3,4], which are not realizable in the
47 laboratory In this paper, we present a non-chaotic system which,
48 we show, could be used to test the different Newtonian and
49 special-relativistic low-speed trajectories
50 Consider the motion of a particle, with rest mass m0and charge
51 q, in a constant uniform magnetic field B, where the initial velocity
52 v of the particle is perpendicular to B According to Newtonian
53 mechanics, the particle moves with a constant linear speedvin a
54 circular path of radius
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rNR¼m0v
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58 The angular speed of the particle is also constant given by
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xNR¼qB
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62 and thus the angular position of the particle varies linearly with
63 time t
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hNRðtÞ ¼ h0þqB
67 According to special-relativistic mechanics, the particle also
68 moves in a circular path with constant linear speed v However,
69 the radius of the circular path is given by[5]
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rR¼ m0v qB
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ðv=cÞ2
72 73 the constant angular speed is given by[5]
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xR¼qB
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ðv=cÞ2
q
77 and thus the angular position of the particle varies linearly with
78 time t as
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hRðtÞ ¼ h0þqB
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ðv=cÞ2
q
m0
82 The only difference between the non-relativistic and relativistic
83 expressions (which are all exact) for the radius, angular speed and
84 angular position is in the mass term – rest mass m0and relativistic
85 mass ffiffiffiffiffiffiffiffiffiffiffiffiffiffim 0
1ðv=cÞ 2
p in, respectively, the former and latter expressions
86
At low speed, wherev c, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðv=cÞ2
1 ðv=cÞ 2
2 , which is
87 close to one The non-relativistic and relativistic radius and angular
88 speed are therefore always close to one another
89
http://dx.doi.org/10.1016/j.rinp.2016.12.035
2211-3797/Ó 2016 Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
E-mail address: lan.boon.leong@monash.edu
Results in Physics xxx (2016) xxx–xxx
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Results in Physics
j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m / r e s u l t s - i n - p h y s i c s
29 December 2016
Please cite this article in press as: Lan BL A proposed test of special-relativistic mechanics at low speed Results Phys (2016),http://dx.doi.org/10.1016/j rinp.2016.12.035
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95 However, the difference between the non-relativistic and
96 relativistic angular position grows linearly with time t
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hNRðtÞ hRðtÞ 1
2
v2
c2
qB
99
100 The time it takes for the difference to grow toDis given by
101
t 2D
ðv=cÞ2
103
104 This time, which increases asv=c decreases, has a power-law
105 dependence on v=c, with exponent 2 As an example,Table 1
106 shows the time it takes for the difference to grow to 0.1 rad
107 (5.7 degree) for differentv=c in the case of a proton in a 0.01 T
108 magnetic field For instance, forv= 104c, the time is 0.348 min,
109 whereas forv= 105c, the time is 34.8 min The relativistic radius
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of the proton’s circular path, which decreases as v=c decreases
111 [see Eq.(4)], is 3.13 cm and 3.13 mm, respectively For comparison,
112 for an electron in a 105T magnetic field withv= 105c, the time is
113 19.0 min and the relativistic radius is 1.71 mm These results
114 suggest that it is possible to test the different predictions of
115 special-relativistic and Newtonian mechanics for the angular
posi-116 tion of a charged particle moving at low speed in a circular path in
117
a constant uniform magnetic field Such a test of special-relativistic
118 mechanics is essentially a test of the relativistic mass formula at
119 low speed (v c) In contrast, previous tests (see references in
120 [6]) of the relativistic mass formula based on the motion of charged
121 particles in electric and magnetic fields were for high speeds
122 ranging from 0.26c to 0.99c (see Table 11.2 in[6])
123 Acknowledgement
124 This work was funded by a Fundamental Research Grant
125 FRGS/1/2013/ST02/MUSM/02/1
126 References
127 [1] Liang SN, Lan BL PLoS ONE 2012;7(4):e34720.
128 [2] Liang SN, Lan BL Res Phys 2014;4:187–8.
129 [3] Lan BL, Borondo F Phys Rev E 2011;83:036201.
130 [4] Lan BL Chaos 2006;16:033107.
131 [5] Barton G Introduction to the relativity principle West Sussex: John Wiley &
132 Sons; 1999.
133 [6] Zhang YZ Special relativity and its experimental foundations Singapore: World
134 Scientific; 1997.
135
Table 1
Time it takes for the difference between the non-relativistic and relativistic angular
position of a proton, which moves in a circular path in a constant uniform magnetic
field of 0.01 T, to grow to 0.1 rad for different ratiov=c The relativistic radius of the
circular path is also given in the last column.
29 December 2016
Please cite this article in press as: Lan BL A proposed test of special-relativistic mechanics at low speed Results Phys (2016),http://dx.doi.org/10.1016/j rinp.2016.12.035