On the classical dynamics of charged particle in special class of spatially non-uniform magnetic field. Ranveer Kumar Singh∗ Department of Physics Indian Institute of Science Education and Research Bhopal Bypass road,Bhauri Bhopal Madhya Pradesh,India. Pin 462066. E-mail:
[email protected] Abstract Motion of a charged particle in uniform magnetic field has been studied in detail classically as well as quantum mechanically.But classical dynamics of a charged particle in non-uniform magnetic field is solvable only for some specific cases.We present in this paper a general integral equation for some specific class of non-uniform magnetic field and solutions of some of them.
1.Introduction It is known that a charged particle in a uniform magnetic field exhibits circular motion[1] with specified frequency of revolution called cyclotron frequency and given radius which depends on the charge and mass of particle;and magnetic field.Quantum mechanically a charged particle in uniform magnetic field has quantised energy levels called landau levels[2]. 1
In non-uniform magnetic field of different kind we have different effects on dynamics of particle classically depending on type of non-uniformity.Some of them are Grad B drift,Curvature drift[3] etc.There are also applications of these effects like magnetic mirrors which are used to confine plasmas[4].The motion of an electron in a magnetic field of constant gradient has been analysed by Seymour(1959)[5] where he has derived the x and y coordinates of electron in terms of elliptic integrals.The non-uniform magnetic field required for an electron to exhibit trochoidal trajectory has been calculated and it turns out to be a fuction of x[6].The quantum mechanical treatment of charged particle in a class of non-uniform magnetic field has been studied using Isospectral Hamiltonian and Supersymmetric quantum mechanics[7].This analysis gives the same Landau Level spectrum[8].Although classical trajectory of a charged particle cannot be solved for all types of general non-uniform magnetic field but we can indeed solve some classes of non uniform magnetic field.It should be noted that we are only considering spatially varying magnetic field and not time varying.We use Landau guage to restrict vector potential to just one component for constant magnetic field along z direction and then introduce some function in vector potential which leads to nonuniform magnetic field.Using elementary classical mechanics we obtain an integral equation which can be solved to get time evolution of x and y coordinate.
2.Charged particle in uniform magnetic field. First we solve by quadrature ;the trajectory of a charged particle in a uniform magnetic field which is known to be circular.Let us assume constant magnetic field in z direction i,e ˆ ~ = B k.Vector B potential for constant magnetic field is given by: ~ ~ = − 1 (~r × B) A 2
(1)
Direct calculation for the magnetic field above gives Ax = −yB/2 and Ay = xB/2 and the z component is Zero.We can choose Landau guage to reduce A to one component such that 2
either Ax = −yB or Ay = xB.Let’s take Ax = −yB.Thus lagranzian for the system can be written as:
L=
1 2 i ( 2 mq˙i
P
+ qA.q˙i )
where symbols have there usual meaning.Note that we can assume no motion in z direction.So expanding the lagranzian gives: 1 L = m(x˙ 2 + y˙ 2 ) − qyB x˙ 2 x is a cyclic coordinate so px =
∂L ∂ x˙
is conserved[9].Thus we have px =
(2)
∂L ∂ x˙
= mx˙ − qBy = C
or x˙ = k1 + k2 y
(3)
where k1 = C/m;k2 = qB/m.Since lagranzian has no explicit time dependence so Hamiltonian of the system is also a constant of motion.We can get the Hamiltonian using :
H=
X (pi q˙i − L)
(4)
i
where pi =
∂L .Direct ∂ q˙i
calculation gives: 1 H = m(x˙ 2 + y˙ 2 ) 2
(5)
q y˙ = ± k3 − k12 − k22 y 2 − 2k1 k2 y
(6)
(3) and (4) gives :
where k3 = 2H/m.Thus we have : dy ±p = dt α + βy + γy 2
3
(7)
where α = k3 − k12 ; β = −2k1 k2 ; γ = −k22 .Integrating both sides with appropriate limits we get(one can check in integral table):
√
1 √ cos−1 [−(β + 2γy)/ q] = ±t −γ
(8)
with q = β 2 − 4αγ.Putting all the values and simplifying gives : √ y=
k3 k1 cos k2 t − k2 k2
(9)
From (3) we can integrate and calculate x to get : √ x=
k3 sin k2 t k2
(10)
Equation 8 and 9 define a circle (as can be checked easily by squaring and adding) with defined frequency of revolution called cyclotron frequency ω = k2 = qB/m.We can also √
calculate the radius of the orbit given by r =
k3 k2
= mv/qB by noting that if energy of
the system is conserved then H = mv 2 /2 where v is the initial velocity of the particle.Thus we have analysed the constant magnetic field case using quadrature and obtained expected results.
3.Charged particle in a non-uniform magnetic field. We now treat the general case of non-uniform magnetic field.Let us introduce a function depending on y in the Landau gauge as Ax = −yBf (y) which gives us non-uniform magnetic field which can be calculated easily using B = ∇ × A.We get :
~ = (yBf 0 (y) + Bf (y))kˆ B
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(11)
where f 0 (y) =
df (y) .(11) dy
represents a special kind non-uniform magnetic field.If we specify
f (y) we obtain different classes of non-uniform magnetic fields.We can again assume no motion along z axis. Lagranzian for this case can be written as : 1 L = m(x˙ 2 + y˙ 2 ) − qyBf (y)x˙ 2 Considering symmetry of lagranzian along x we have px =
∂L ∂ x˙
(12)
= mx˙ − qByf (y) = C or :
x˙ = k1 + k2 yf (y)
(13)
where k1 = C/m and k2 = qB/m.Hamiltonian for the system can be calculated using (4) and simple calculation gives (5).Hamiltonian is again a constant of motion as lagranzian has no explicit time dependence.(5) and (13) gives : q y˙ = ± k3 − k12 − k22 y 2 f 2 (y) − 2k1 k2 yf (y)
(14)
(14) gives us the desired integral equation which can be solved for different f (y) and in turn for different classes of non-uniform magnetic field to get y as a function of t.This y can be substituted in principle in (13) to get x as a function of t.These two equations define the trajectory of the particle and other parameters of motion.The integral equation is : Z
Z
dy p
k3 − k12 − 2k1 k2 yf (y) − k22 y 2 f 2 (y)
=±
dt + K
where K is the constant of integration.
3.1. Special Cases We can check that (15) yields correct results for some special cases. Case 1. f (y)=1
5
(15)
In this case we get constant magnetic field along z direction as can be checked by putting f (y) = 1 in (11).Putting f (y) = 1 in (15) gives back (7) which we have already solved.Thus we get circular trajectory for constant magnetic field. Case 2. f (y) = 1/y ~ = 0.This means particle must go undeviated i,e the For this case we can calculate that B trajectory should be straight line.Putting f (y) = 1 Eq.(15) gives : √
R
dy k3 −k12 −2k1 k2 −k22
= ±t + K
The denominator is just a constant (say a).So we get: y = at + K 0 where K 0 = aK It can be noted that both the signs + and − gives the same trajectory.We can calculate x from (13) which comes out to be: x = bt + D where b = k1 + k2 and D is constant of integration. x and y indeed define a straight line.We can check that x˙ 2 + y˙ 2 = v 2 where v is the initial velocity of the particle. 3.2. Note: We can extent the same procedure for a magnetic field non-uniform in x coordinate.In that case we use the Landau guage in which Ax = 0 = Az and Ay = xBf (x).Magnetic field in this case takes the form: ~ = (xBf 0 (x) + Bf (x))kˆ B
(16)
Accordingly lagranzian assumes the form given by: 1 L = m(x˙ 2 + y˙ 2 ) + qxBf (x)x˙ 2 Now y is a cyclic coordinate so that py =
∂L ∂ y˙
(17)
= mx˙ + qBxf (x) = C or :
y˙ = k1 − k2 xf (x) 6
(18)
where k1 = C/m and k2 = qB/m.Hamiltonian of the system remains the same.Invoking energy conservation and using (17) and (5) we obtain : q x˙ = ± k3 − k12 − k22 x2 f 2 (x) + 2k1 k2 xf (x)
(19)
This gives us another integral equation which can be solved to obtain particle’s x coordinate as a function of time which can be substituted in (17) to obtain particle’s y coordinate.Thus we can obtain particle’s trajectory and other parameters of motion.The integral equation is given as : Z
Z
dx p
k3 − k12 + 2k1 k2 xf (x) − k22 x2 f 2 (x)
=±
dt + K
(20)
where K is the constant of integration. Special cases above gives us the same result as it should be.
4. Exponentially decaying Magnetic field Although (15) can be solved for different kinds of non uniform magnetic fields,but Let’s consider an exponentially decaying magnetic field.Let the magnetic field be given by :
~ = Be−y kˆ B
(21)
where B is a constant.From (11) we can solve for f (y).Thus we have : Be−y = yBf 0 (y) + Bf (y) This is an ordinary differential equation. Solution to this is given by:
f (y) =
c e−y − y y
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(22)
we can fix c = 1 so that, 1 f (y) = (1 − e−y ) y
(23)
magnetic field still remains the same.Putting this in (15) gives : Z
Z
dy
p =± k3 − k12 − 2k1 k2 (1 − e−y ) − k22 (1 − e−y )2
dt + K
(24)
let’s put a = k3 − k12 ; b = k22 ; c = 2k1 k2 then the integral has the solution given by ; arcsin
2(c+b−a)ey −c−2b √ c2 +4ab
√ ( c + b − a)
+ K 0 = ±t + K
(25)
to make equations look simpler let’s put c + b − a = α2 ; c + 2b = β and by calculation √ c2 + 4ab = 2k2 k3 .constants of integration can be manipulated such that K 0 = K so the solution after rearranging terms looks as : √ β k3 k2 y = log sin(±αt) + 2 α2 2α
(26)
From (13) we can calculate x as : Z x= k1 + k2 − √
where l =
k3 k2 ;m α2
=
β .Solution 2α2
k2 dt l sin(±αt) + m
(27)
of this equation is :
±l m tan( αt 2 ) √ m2 −l2
2 arctan √ x = (k1 + k2 )t − k2 α m2 − l 2
(28)
Simple calculation gives :
x = (k1 + k2 )t − 2 arctan
8
! (m/l) tan αt ± 1 2 p 2 (m/l) − 1
(29)
in terms of original constants we have :
x = (k1 + k2 )t − 2 arctan
√ ! ± k3 (k1 + k2 ) tan αt 2 p (k1 + k2 )2 − k3
(30)
Equations (26) and (30) define the trajectory of the particle. Note : If we have a particle with zero energy in magnetic field then it will reamin stationary since magnetic field does no work.In this case this is indeed true.If we take zero energy then k3 = 0 and a bit of calculation shows that x = 0 and y=log(k2 )=constant.Thus the particle remains stationary on point (0,log(k2 )). The magnetic field is shown in figure below.
~ = Be−y kˆ Figure 1: Exponentially decaying magnetic field B
field.png
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Now to analyse particle’s trajectory in this field let’s take assume the following : k3 = 1; k2 = 0.1(we have taken specific charge of the particle to be 1 and assumed a strong magnetic field of 0.1T!);k1 + k2 = 1.118(just for making things simple!)This gives us : x = 1.12t − 2 tan−1 (2.24 tan(t/4) + 2); y = log(0.08 sin(t/2) + 0.1) We plot these parametric equation for t ∈ (0, 50).
~ = Be−y kˆ Figure 2: Particle’s trajectory in exponentially decaying magnetic field B
Thus the particle shows periodic motion as can be inferred from the trajectory.There are several forms of f (y) for which solution to (15) exists and thus such class of non-uniform magnetic field can be analyzed easily. 10
5. Conclusion Here we have presented a method to analyse motion of charged particle in various classes of non-uniform magnetic field.Although the integral equation presented do not have trivial solution for many forms of f (y) or equivalently f (x) but for such cases numerical integration can be used to find out the trajectory.As shown many forms have exact solution and thus it is useful to further study this method and generalise it to other forms.
Acknowledgement This work was carried out at Panjab University Chandigarh.The author is indebted to Prof. C.N Kumar whose guidance helped to complete this work.The author also thanks Ms. Harneet Kaur,Dr. Amit Goyal for useful discussions.
References (1,9)Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addisonwesley. (2)Landau, L. D.; and Lifschitz, E. M.; (1977). Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics. Vol. 3 (3rd ed. London: Pergamon Press). ISBN 0750635398 (3)Baumjohann, Wolfgang; Treumann, Rudolf (1997). Basic Space Plasma Physics. ISBN 978-1-86094-079-8. (4) Principals of Plasma Physics, N Krall, 1973, Page 267 (5)Seymour,P.W-Drift of a charged particle in magnetic field of constant gradient.Aust.Jour.Phys 12:309-14 (6)RF Mathams,Motion of a Charged Particle in a Non-uniform Magnetic Field,Australian
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Journal of Physics 17(4) 547 - 552 (7)F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in quantum mechanics, Phys. Rep. 251, 267-285, (1995) (8)A. Khare, C.N. Kumar, Landau Level Spectrum for Charged particle in a Class of NonUniform Magnetic Fields , Mod. Phys. Lett. A 8, 523-529 (1993)
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