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Average Lorentz self-force from electric field lines
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European Journal of Physics Eur. J. Phys. 36 (2015) 055012 (11pp)
doi:10.1088/0143-0807/36/5/055012
Average Lorentz self-force from electric �eld lines Sandeep Aashish and Asrarul Haque Department of Physics, BITS Pilani Hyderabad Campus, Hyderabad-500078, AP, India E-mail:
[email protected] and
[email protected] Received 1 January 2015, revised 17 May 2015 Accepted for publication 28 May 2015 Published 26 June 2015
Abstract
We generalize the derivation of electromagnetic �elds of a charged particle moving with a constant acceleration Singal (2011 Am. J. Phys. 79 1036) to a variable acceleration (piecewise constants) over a small �nite time interval using Coulomb s law, relativistic transformations of electromagnetic � elds and Thomson s construction Thomson (1904 Electricity and Matter (New York: Charles Scribners) ch 3). We derive the average Lorentz self-force for a charged particle in arbitrary non-relativistic motion via averaging the �elds at retarded time. ʼ
ʼ
Keywords: electric
�eld
ʼ
lines, Thomson s construction, self-force
1. Introduction The electromagnetic �elds [1] of a charged particle moving with a constant acceleration are obtained exploiting Coulomb force, relativistic transformations of electromagnetic �elds and Thomson s construction [2]. The derivation of the �elds for an accelerated charge is carried out in the instantaneous rest frame. The geometry of the Thomson s construction makes it evident that �elds pick up the transverse components proportional to acceleration in addition to the radial components. The electromagnetic �elds so obtained turn out exactly the same as those obtained via Lienard –Weichert potentials. This method is mathematically simpler than the usual method [3] of computing the electromagnetic �elds which involves rather cumbersome calculations. However, the one downside to this method is that it is not suf �cient to calculate the radiation reaction force. The derivation of radiation reaction involves non-uniform acceleration of the charged particle. In this paper, we address the question as to how to calculate the EM �elds for a charged particle moving with a variable acceleration (piecewise constants) over a small �nite time ⃗ and B ⃗, Coulomb �eld of a stationary interval Δt using relativistic transformations of �elds E ʼ
ʼ
0143-0807/15/055012+11$33.00 © 2015 IOP Publishing Ltd Printed in the UK
1
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque ʼ
charge as well as Thomson s construction. Consider a charge moving with piecewise different constant accelerations in time interval Δt . The charge moving with piecewise N (say) different constant accelerations over time interval Δt enables us to use the relativistic transformations ⃗ and B ⃗ of a uniformly moving charge through N small time sub-intervals Δt N of �elds E (say). The electromagnetic �elds at some space and time points are obtained by time-averaging out the �elds stemming from the piecewise N different constantly accelerated motions of the charge through N temporal sub-intervals Δt N . A charged particle moving with non-uniform acceleration radiates. A radiating charged particle experiences a force which acts on the charge particle and is called as self-force. The Lorentz self-force [3] (p 753) arising due to a point charge conceived as a uniformly charged spherical shell of radius s is given by
self F ⃗ =
−
2
q2
2
q2
2
v˙ ⃗ (t )
+
in the
�rst
term stands for electromagnetic mass and becomes
3 4 πε0 c s
3 4 πε0 c
3
v¨⃗ (t )
+ O (s)
with
s⃗
= s,
(1)
where, • the
quantity
2
q2
3 4 πε0 c 2 s
divergent as s → 0+ , • the second term represents the radiation reaction and is independent of the dimension of the charge distribution and • the third term corresponds to the �rst �nite size correction and is proportional to the radius of the shell s . It is plausible to expect that the piecewise N different constantly accelerated motions of the charge through N temporal sub-intervals Δt N could give rise to the average self-force. We derive the average Lorentz self-force for the charged particle in arbitrary non-relativistic motion via averaging the said retarded �elds [4].
2. Preliminary In this section, we shall brie �y discuss about the relativistic transformations of the �elds and Thomson s construction [2]. We shall further discuss and summarize the results on the electromagnetic � eld of a constantly accelerated charge in the instantaneous rest frame [1] and self-force [4]. ʼ
!
!
2.1. Relativistic transformations of E and B of a uniformly moving charge
Let us consider two frames S and S ′. Let S ′ is moving with constant velocity v ⃗ = β ⃗c relative to S . Suppose a particle of charge q moves with a velocity v ⃗ relative to S . The charged particle would thus appear to be at rest with respect to the system S ′. ⃗ and magnetic � elds [6] B ⃗ of the charged particle in frame S is related to the The electric E electric E and magnetic �elds B ′⃗ of the charged particle in the frame S ′ as follows: ′ ⃗
E ⃗ = E ′⃗ ∥
+ γ ⎡⎣ E ′⃗ ⊥ − β ⃗ × B ′⃗ ⎤⎦, B ⃗ = B ′⃗ ∥ + γ ⎡⎣ B ′⃗ ⊥ + β ⃗ × E ′⃗ ⎤⎦
E ′⃗ = E∥⃗ + γ ⎡ ⎣ E⊥⃗
+ β ⃗ × B ⃗⎤⎦, B ′⃗ = B∥⃗ + γ ⎡⎣ B⊥⃗ − β ⃗ × E ⃗⎤⎦ .
In case, the charged particle moves with non-relativistic speed ∣ β ⃗ ∣ �eld transformations simplify to yield: 2
<< 1 i.e. γ →
(2)
⃗ and B ⃗ 1, E
Eur. J. Phys. 36 (2015) 055012
E ⃗ = E ′⃗
S Aashish and A Haque
− β ⃗ × B ′⃗ ,
B ⃗ = B ′⃗
+ β ⃗ × E ′⃗
(3)
E ′⃗ = E ⃗ + β ⃗ × B ⃗, B ′⃗ = B ⃗ − β ⃗ × E ⃗.
In S ′ frame, Therefore,
�eld
B ′⃗
(4)
is purely electric as the charge is at rest with respect to the system S ′.
= 0, E ⃗ = E ′⃗ and B ⃗ = β ⃗ × E ′⃗ .
(5)
Now, suppose that the charged particle is moving along the Z -axis (θ = 0 ) i.e. β ⃗ = β zˆ , then in the spherical polar coordinates (R, θ, ϕ) , we have E ⃗ = E R Rˆ E ′⃗
=
e2 ˆ R , R 2
= E R′ Rˆ ′ =
B ⃗ = β ⃗ × E ′⃗
(6)
e2 ˆ R′ , R′2
= β ⃗ × E ⃗ =
(7) Bϕ ϕˆ
=
eβ sin θ ˆ ϕ. R 2
(8)
′⃗ implies that R = R′ where R is the distance between the Now, E ⃗ = E location of the charge in the frame S .
�eld
point and the
2.2. Thomson s construction ʼ
Consider a charged particle, initially moving with a constant velocity v I , suffers a change in velocity after the time interval (0, τ ) to a constant velocity v F ⃗ . Suppose the charged particle undergoes an acceleration to a small velocity Δv ( Δv << c ) for the short time τ . Arguments due to Thompson, regarding the resulting �eld distribution in terms of the electric �eld lines after time t = T , attached to the accelerated charge are summarized as follows: ⃗
⃗
any time t < 0, �elds are that of the charge moving with constant velocity v I ⃗ . The electric �eld lines will emanate radially outward from the charge in all possible directions. The information pertaining to the change in motion (acceleration) can ’t reach outside a sphere of radius R = cT . • For any time 0 < t < τ , � elds are that of the charge undergoing acceleration. The electric �eld lines will admit distortions in the form of a kink in a region between the two spheres S 1 and S 2(as shown in � gure 1 ) in order to preserve the continuity of the � eld lines. Thus, the �elds would now begin to pick up the tangential component in addition to the radial one. The information pertaining to the change in motion is con �ned in the spatial region cT < R < c (T − τ ). • For any time t > τ �elds are that of the charge moving with constant velocity vF . The electric �eld lines will emanate radially outward from the charge in all possible directions. The information pertaining to the change in motion can ’t reach inside a sphere of radius R = c (T − τ ). • For
⃗
2.3. Electromagnetic field of the constantly accelerated charge [1]
Consider a charged particle moving in a lab-frame S . Suppose the charge is moving with a constant initial velocity v 0⃗ . Let the charge be uniformly accelerated ⃗ for a short time interval (− Δt 2, Δt 2) to a velocity Δv ⃗ (Δv << c) so that its velocity become v 1⃗ (constant). The velocity of the charge at t = 0 is v ⃗ = (v0⃗ + v1⃗ ) 2 . 3
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
ʼ
Figure 1. Electric �eld line as per Thomson s construction exhibiting the ‘kink ’ in
between the spherical surfaces S 1 and S 2 corresponding to the acceleration of the charge.
Consider a (instantaneous rest) frame S ′ moving with velocity v ⃗ = (v0⃗ + v1⃗ ) 2 relative to frame S . The initial and �nal velocities say v 0 and v ′ 1⃗ respectively of the charged particle relative to frame S ′ turn out equal and opposite v ′⃗ 1 = −v ′⃗ 0 ≡ v ′⃗ (say). For convenience, S ′ could be rotated (rotation can be undone at the end) so that the charge motion is along the horizontal axis. Suppose the charge be instantaneously at rest at O′ at t ′ = 0 . During the time of acceleration Δt ′, the charge moves a distance ′ Δt ′2 8 towards O′ for time Δt 2 and then turns back for the remaining time. To the �rst order in Δt ′ , charge could be assumed to be practically at rest at O′. Consider the � elds of the charge at time T ′ >> Δt ′. Let O1 and O2′ (for N = 1) be the positions of the charge at T ′ − Δt ′ 2 and T ′ + Δt ′ 2 respectively. The electric �eld in the regions (T ′ + Δt′ 2) < r′ < c (T ′ − Δ t′ 2) would be in the radial direction from ′ the points O1 and O2 . We wish to calculate the electric �eld in the region c (T ′ − Δt′ 2) < Δr′ < c (T ′ + Δ t′ 2) which possesses the information of the change in motion of the charge. Geometrically, it is obvious from the Thomson s construction (for N = 1 as shown in �gure 1), that the � eld now picks up both the radial ( E r ) as well as the transverse components ( E θ ′′) both at A and P . The spatial variation in the transverse components of electric �eld over a distance Δr ′ from A to P turns out, ′⃗
′
′
′
ʼ
′
′
∂ E θ ′′ −e β˙′ sin θ ′ = . (9) ∂r ′ c r ′2 The formal solution of (9) at P (r′, t ) assuming that �eld falls to zero as r ′ → ∞ leads to e β˙′ sin θ ′ . (10) E θ ′′ = cr ′
4
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
Figure 2. A schematic of the motion of a piecewise constantly accelerated charge. The
particle moves with different constant velocities for times (1 − ε ) Δt
constant accelerations for times
N
εΔt N
as well as with different
. The shallow curved paths belong to the
accelerated regions.
The total electric
�eld
′⃗ = E
at P (r′ , t ) could be written as:
q ˆ r ′ r ′2
+
q
r ′⃗
(
× r ′⃗ × v ˙′⃗
c2
r ′
).
(11)
The transformation of the �eld from S ′ to S yields: r ˆ − β ⃗
E ⃗ = q
2 2
r γ
3 ( 1 − r ˆ · β ⃗)
In the non-relativistic case (
v c
r
2
3 ( 1 − r ˆ · β ⃗)
+
q
×
rˆ
rˆ
{ ( r ˆ − β ⃗) × β ˙⃗} .
3 r ( 1 − r ˆ · β ⃗)
c
<< 1 i.e. γ →
ˆ − β ⃗ r
E ⃗ = q
+
q
(12)
⃗ takes the form: 1), the expression for E
{ ( r ˆ − β ⃗) × β ˙⃗} .
×
3 ˆ · β ⃗) r ( 1 − r
c
(13)
2.4. Results of the self-force [4]
A simple derivation of the self-force [4] based on the consideration that the averaged value of the �eld in the suitably small closed region surrounding the point charge is the value of the �eld under consideration at the position of the point charge is carried out in detail. The selfforce is de�ned as: F Self ⃗ ( r ⃗, t )
= q Lim+ E ⃗ ( r ⃗, t ) = s →0
⃗ ( r ⃗, t ) , qESelf
(14)
where E ⃗ (r ⃗, t )) is average �eld over the surface of a spherical shell of radius s and the �eld E ⃗ (r ⃗, t ) depends upon the position and motion of the charge particle at the retarded time. Using the �eld due to an accelerated charged particle (in the limit v c → 0), the self force turns out: 5
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
Figure 3. Motion of a piecewise constantly accelerated charge as observed in the instantaneous rest frames S i′. The charge moves with initial velocity −vi′⃗ and �nal
velocity
v ′ in i⃗
F Self ⃗ ( r ⃗, t )
the frame
=−
S i′.
q2
⎛ a ⃗ (t − s c) ⎞ ⎜ Lim ⎟. ⎠ 3 4 πϵ0 c 2 ⎝ s →0 + s
2
3. Calculation of average electric
(15)
�eld
Consider a charge moving with an initial velocity v 0⃗ in the lab frame S (as shown in � gure 2). Suppose it undergoes accelerations from − Δt 2 to Δt 2. We consider that the acceleration in the time interval Δt is not continuous but rather consists of a series of a �nite number of different piecewise constant accelerations. Let us divide the total time interval Δt into a large number 2N (N ⩾ 2) of sub-intervals. Suppose all the odd and even sub-intervals are of lengths (1 − ε ) Δt N and εΔt N respectively. We consider that the charge undergoes non-zero constant accelerations in the odd sub-intervals accompanied by non-zero constant velocities in the even sub-intervals. Accelerations {a i⃗ (τ i ) : i = 1, 3, 5, … , 2N − 1} and velocities {v i⃗ (t ) : i = 0, 2, …, 2N} are de�ned as:
⎧ i⃗ θ ( τi − ti −1 ) θ ( ti − τi ); i = 1, 3, 5, …, ⎪ a i⃗ ( τ i ) = ⎨ 0, ( ti < τ i < ti ); i = 0, 2, …, 2 N
+1
⎪
⎩
v i⃗ + 1 (t )
2N
= vi⃗ +
Δt 2
∫ −
Δt 2
ai⃗ +1 ( τi )dτ i; i
= 0, 2, …, 2N,
− 1
(16)
(17)
where
⎧⎛ ⎜− ⎪ ⎪⎝ t i = ⎨ ⎪ ⎛⎜ − ⎪ ⎩⎝
1 2 1 2
+ +
i ⎞ ⎟ Δt ; i 2 N ⎠ i
= 0, 2, …, 2 N .
+ 1 − 2ε ⎞⎟ 2 N
Δt ; i
⎠
= 1, 3, 5, …, 2N − 1.
The charge undergoes through various different piecewise constant accelerations in the time interval Δt . In order to determine the EM �elds of an accelerated charge over Δt , we require as many instantaneous rest-frames as that of the different constant accelerations. 6
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
ʼ
Figure 4. Electric �eld line O3FGHI as per Thomson s construction exhibiting the
kink ’ in between the innermost and the outermost spherical surfaces corresponding to the accelerated motion of the charge. The thin annular region between the spherical surfaces belongs to the non-accelerated motion of the charge. ‘
These instantaneous rest-frames could be obtained by appropriate boosts. Let us consider the ith instantaneous rest-frame {Si′ : i = 1, 3, 5, …, 2N − 1}(as shown in �gure 3) moving with velocity {v S⃗ i′ = (vi⃗ + vi⃗ +1 ) 2 : i = 1, 3, 5, …, 2 N − 1 }. We assume that ∣ vi⃗ +1 − vi⃗ ∣ << c. Suppose the charged particle appears to be instantaneuosly at rest at ti′ + t i′+1 ( ; i = 0, 2, …, 2N − 2) where 2 ⎧⎛ 1 i ⎞ ⎜− ⎟ Δt ′ ; i = 0, 2, …, 2 N . + ⎪ ⎪⎝ 2 2 N ⎠ t i′ = ⎨ ⎪ ⎛⎜ − 1 + i + 1 − 2ε ⎞⎟ Δt′ ; i = 1, 3, 5, …, 2N − 1. ⎪ ⎠ 2 N ⎩⎝ 2 ʼ
Thomson s construction for N = 2 is shown in the � gure 4. In � gure 4, points B 1 and A 1 stand for the begining and end of accelerations with regards to theinstantanous rest frames 1′. The electric � elds along O2′ B2 and O3′ A2 are inclined at small angle Δθ 1′ with respect to O′P . In the frame S i′, the corresponding transformed velocities v i′⃗ −1 and v i′⃗ +1 are given by: v i′⃗ +1
= −vi′⃗ −1 = vi′⃗ (say).
(18)
The acceleration in S i′ reads:
i⃗ ′ =
2vi′⃗ (1
− ε ) Δt ′
N
.
(19)
Without any loss of generality, we take the orientation of S i′ such that motion happens along Z i′. The calculation of E i′ in i′ proceeds in a similar way to that in the section 2.3. At the point 7
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
Figure A1. Piecewise constant accelerations having smooth change at the temporal
boundaries.
P, the charge is instantaneously at rest with respect to the origin O ′. The electric � eld E i′ at a later time T ′ >> Δt′ N at P is obtained as:
i⃗ ′ = E
q r ′2
nˆ′
+
q nˆ′
× nˆ′ × β ˙i⃗ ′
c
r
.
In the above expression, we have made use of Oi′ Bi
≈
Oi′+1 Ai
≈ O ′P = r ′ ;
as Δθ ′ is small. Transformation of ( β i⃗ → 0, γ → 1) yields:
i⃗ E
=
q r 2
nˆ
+
q nˆ c
′ E i⃗
N
to the lab frame S for the non-relativistic velocity
× nˆ × β ˙i⃗ r
= 1, 2, …,
i
,
r ⃗ ˙i⃗ = a i⃗ ( τ i ) . The quantities r and β ˙i⃗ on the right hand side are evaluated at where, ˆ = and β r c retarded time, t Ri = τ i − r c . The electric �eld at P
⃗
⎧ ˙⃗ ⎪ q nˆ + q nˆ × nˆ × β i ; i = 1, 3, …, 2N − 1 ⎪ i⃗ = ⎨ r 2 c r E ⎪ q nˆ; i = 0, 2, … , 2 N . ⎪ ⎩ r 2
In fact consists of N number of piecewise different values of E i belonging to 2N − 1 subintervals over Δt . It is therefore plausible to consider the electric �eld at P in the vicinity of a point in time as the time-averaged value 〈E 〉⃗ of the electric � elds E i for the entire time Δt . The time-averaged electric �eld is obtained as: ⃗
8
⃗
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque Δt 2
⃗ E
=
∫ − ∑ = E ⃗ dt ∑ = ⎛⎝ E ⃗ − 1 N − ε Δt + E ⃗ N ε Δt ⎞⎠ N
N
i
i 1
Δt 2
Δt 2
∫ −
Δt 2
i 1
=
⎜
2i 1
⎟
2i
Δt
dt
˙ ˙ ⎛ ⎞ N q q nˆ × nˆ × β 2⃗ i − 1 ⎟ q nˆ × nˆ × β 2⃗ i − 1 ε ⎜ = ∑ 2 nˆ + − . ⎟ N ∑ c N i =1 ⎜ r c r r i =1 ⎝ ⎠ N
1
For ε
<<
Δt , we have: N
⃗ E
=
q 2
r
nˆ
+
q
nˆ
× nˆ ×
c
(20)
˙ β ⃗ ,
r
(21)
where 1
˙ ⃗ β
=
Δt 2
∫ c −
Δt 2
=
Δt
∑ =1 a 2⃗ − 1 c 2 − 1 N
a 2⃗ i − 1 ( τ2 i −1 )dτ 2i −1
( τ i
i
)
i
.
N
(22)
4. Calculation of the self force The self force of a charge moving with arbitrary velocity, in general, contains acceleration and higher derivatives of acceleration as is especially obvious from equation ( 1). A charge moving q2 ¨ 2 with constant acceleration does not experience any radiation reaction as the term v ⃗ (t ) 3 4πϵ0 c 3 vanishes. In the case at hand, charge is moving with non-zero constant acceleration in the time interval (1 − ε ) Δt N whereas with zero acceleration in the time interval εΔt N . Therefore, the charge con�ned to these time intervals will not experience any radiation reaction force. However, over the interval Δt , the charge moving with various different constant accelerations would give rise to a net change in the acceleration Δa ⃗ over Δt . This suggests that over the time Δt , Δa ⃗ Δt is no longer zero, and hence the charge must experience average radiation reaction. The average self-force [4] may be de�ned as 2 F Self ⃗ ( r ⃗, t ) = −
q2
Lim
a ⃗ ( τ 2i −1
3 4 πε0 c 2 s →0 +
− s c)
s
,
(23)
where a 2⃗ i −1 ( τ2i −1 − s c)
= a2⃗ i −1 θ ( τ2 i −1 − t2 i −2 − s c) θ ( t2 i −1 − τ2 i −1 + s c) . We can Taylor expand a 2⃗ i − 1 (τ 2 i−1 − s c) about c so that, a 2⃗ i −1 ( τ2i −1 − s c)
= a2⃗ i −1 ( τ 2i −1 ) −
s
d
c dτ 2i −1
a 2⃗ i − 1( τ 2i − 1)
+ O ( s 2)
The self force expression now becomes F Self ⃗ ( r ⃗, t )
=−
2
q2
Lim
a ⃗ ( τ 2i − 1 )
3 4 πε0 c 2 s →0 +
s
9
+
2
q2
3 4 πε0 c 3
a˙ ⃗ ( τ 2i − 1 ) .
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
Since, θ ( τ2i −1 − t2i −2
− s c) = θ ( τ2i −1 − t2i −2 ) −
θ ( t2i − 1 − τ2i −1 + s c ) = θ ( t2i −1 − τ2i −1 )
+
s c s c
δ ( τ2 i −1 − t2 i −2 ) δ ( t 2i −1
+ O ( s 2)
− τ2i −1 ).
Therefore, N
N
∑a ⃗
2i − 1
( τ2i − 1 − s c) = ∑a⃗2i − 1θ ( τ2i − 1 − t2i − 2 ) θ ( t 2i − 1 − τ 2i − 1)
i =1
+
s
i =1 N
∑a ⃗ c
2i − 1 θ ( t2i − 1
−
t2i − 2 ) [ δ ( t 2i − 1 − τ 2i − 1)
−
( )
δ ( τ 2i − 1 − t 2i − 2 ) ] + O s2
i=1
It is evident that 2⃗ i − 1 (τ 2 i −1
−s
c ) turns out divergent at the temporal boundaries:
τ2i − 1 = t2i −1 or τ 2i −1 = t 2i −2 .
However, the time-averaged self force would render a 2⃗ i − 1 (τ 2i −1 − s c) physically sensible. Moreover, in order to prevent this meaningless results, we assume that the transitions from non-zero constant acceleration to zero constant acceleration are smooth at the temporal boundaries (please see appendix for clari�cation). Such sort of meaningless results arise in models that involve step functions forces [5]. Now, q2
2 F Self ⃗ ( r ⃗, t ) = −
a⃗
3 4 πε0 c 2 s
− =−
2
q2
N
Δt 2
1
∫ ∑a ⃗
3 4 πε0 c3 Δt − Δ2t i = 1 q2
2
2
3 4 πε0 c s
a⃗
+
2
2i − 1
[ δ ( t2i −1 − τ2i −1 ) − δ ( τ2i −1 − t 2i −2 ) ] dτ2 i −1
q2
1 3
3 4 πε0 c 2π
a˙ ⃗ ,
(24)
where we have identi �ed a ˙ ⃗
=
a 2⃗ i −1
− a1⃗
Δt
.
Thus, the time-average radiation reaction stems from the time-averaged acceleration.
5. Conclusion We derive the electromagnetic � elds of a charged particle moving with a variable acceleration (piecewise constants) over a small �nite time interval using Coulomb s law, relativistic transformations of �elds and Thomson s construction. We derive the expression for the average Lorentz self-force for a charged particle in arbitrary non-relativistic motion via averaging the retarded �elds. ʼ
ʼ
Appendix A. A model for piecewise constant and smooth acceleration In order to have the physically sensible values of 2⃗ i − 1 (τ 2 i −1 − s c ), we assume that the transition from non-zero constant acceleration to zero acceleration and viceversa is smooth at 10
Eur. J. Phys. 36 (2015) 055012
S Aashish and A Haque
the temporal boundaries. We can incorporate the smooth change in the acceleration at the temporal boundaries by de �ning our acceleration (as shown in �gure A1) as follows:
⎧ a 2i −1, t2i −2 + ε1 ⩽ τi ⩽ t 2i −1 − ε1 ( i = 1, 2, …, N. ) ⎪ ⃗ a 2⃗ i − 1 ⎪ ⎪ a 2⃗ i −1 − 2ε ( τi − t2i − 1 + ε1 ), t2i −1 − ε1 ⩽ τi ⩽ t 2i −1 1 a ⃗ ( τ i ) = ⎨ ⎪ + ε1 (i = 1, 2, …, N − 1. ) ⎪ a 2i −1 ⎪ ⃗ ( τi − t2i − 1 + ε1 ), t2i −2 − ε1 ⩽ τi ⩽ t2i −2 + ε1 (i = 2, 3, …, N ). ⎩ 2ε1 We assume that 2 ε1 τ 0
=
>> τ 0 , where 2 q2 3 mc3
[7]) is de�ned by
0 (see
.
Now,
⎧ a 2⃗ i − 1 , t2i − 1 − ε1 ⩽ τi ⩽ t2i − 1 + ε1(i = 1, 2, …, N − 1) ⎪− ⎪ 2ε1 a ˙ ⃗ ( τ i ) = ⎨ ⎪ a 2⃗ i −1 , t 2i − 2 − ε1 ⩽ τi ⩽ t 2i − 2 + ε1 (i = 2, 3, …, N ). ⎪ ⎩ 2ε1 The quantity
a ˙⃗ now turns out: a ˙ ⃗
=
=
1
Δt 2
N
∫ ∑a˙⃗ ( τ )dτ
Δt − Δ2t i =1
a 2⃗ N − 1
− a1⃗
Δt
i
(A.1)
i
.
(A.2)
References [1] Singal A K 2011 An ab initio derivation of the electromagnetic � elds of a point charge in arbitrary motion Am. J. Phys. 79 1036 [2] Thomson J J 1904 Electricity and Matter (New York: Charles Scribners) ch 3 [3] Jackson J D 2003 Classical Electrodynamics (New York: Wiley) [4] Haque A 2014 A simple derivation of Lorentz self-force Eur. J. Phys. 35 055006 [5] Schwinger J, DeRaad L L, Milton K A and Tsai W 1998 Classical Electrodynamics (New York: Perseus) ch 37 [6] Grif �ths D J 1999 Introduction to Electrodynamics 3rd edn (Upper Saddle River, NJ: Prentice Hall) section 12.2 and 12.3 [7] Moniz E J and Sharp D H 1977 Phys. Rev. D 15 2850 Rohrlich F 2008 Phys. Rev. E 77 046609
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