Поле прецессирующего магнитного диполя Текст научной статьи по специальности «Физика»

UDC 530.1; 539.1

The field of precessing magnetic dipole

V. Epp, M. A. Masterova

Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia

E-mail: epp@tspu.edu.ru

The field of a rotating inclined magnetic dipole is studied. One first integral of motion and some particular solutions of equations of motion for a charged particle in this field are found. The effective potential energy of a charge is studied. It is shown that the effective potential energy in the corotating reference system has six stationary points, which correspond to circular motion of a charge.

Keywords: magnetic dipole, electromagnetic Geld, inclined rotator, charge, equations of motion.

1 Introduction

Magnetic field of the planets and stars can be thought of as a dipole field in good approximation. Motion of a charged particle in the field of magnetic dipole has been investigated in details since 1907. In particular, the Earth magnetic field was studied by St0rmer [1,2]. Solution of equation of motion for charged particle in a dipolar magnetic field gives trapping regions for particles, with energy of limited range. These regions of planets are known as radiation belts [3,4]. The first theoretical studies on the properties of the trajectories of charged particle in a dipolar field are presented in [1,5,6].

There are also well known bodies, for which direction of magnetic moment is different from the direction of axis of rotation. In this case there is an electric field around the body induced by magnetic field. The neutron stars and pulsars are examples of such bodies. Some models of electric field which is generated in the neighbourhood of neutron stars were developed in [7]. All these models are based on suggestion that the neutron star is a conducting body. But there are also celestial bodies, which consist of non-conducting matter and their axis of rotation is inclined with respect to the magnetic field axis. Magnetic field of such objects can be in good approximation described as the field of inclined rotating magnetic dipole.

In this paper we present calculation of the field of rotating magnetic dipole and we consider also equations of motion of a charged particle in this field. It is shown that each stationary point of the effective potential energy corresponds to particular solution of equations of motion for a charged particle.

2 The field of precessing magnetic dipole

Let us consider the field which is produced by precessing magnetic dipole. We define the law of motion of

a dipole moment i in the Cartesian coordinate system as follows:

l = u(sin a cos wt, sin a sin wt, cos a),

(1)

where u > 0 mid w > 0 are the magnitude and angular velocity of the dipole respectively, 0 < a < n is the angle between vector i and the rotational axis. The general formulae for the field of an arbitrary changing dipole are given, for example, in [8]:

E

(n x l) (n X fl)

o + o ,

(n X (n x 1)) 3n(n/i) — /1

H = --------^---------1---------t,------

+

rc2

3n(ni) — i

r3 ,

(2)

(3)

where c is the speed of 1 ight, n = r/r is the unit vector, r is the radius-vector. Field is calculated at time t, and all the quantities on the right side of these equations

r

should be taken at the retarded time t' = t--------.

c

In a spherical coordinate system (r, 0, y>) the electric field vector has the form:

r3Er r3 Eg r3Ev

where

= 0,

= up sin a (p sin t — cos t),

= — up cos 0 sin a (p cos t + sin t)

t = wt — y>, p = —.

c

Magnetic field vector has components:

r3Hr = 2yU, [sin a sin 0(cos t — p sin t)

+ cos 0 cos a],

—u [cos 0 sin a(cos t — psin t — p2 cos t ) — sin 0 cos a] ,

—U sin a(sin t + p cos t —p2 sin t ).

r3Hg

r3H

(4)

(5)

(6)

(7)

(8)

(9)

One can see that time dependence appears only as composition wt' — f. It means that the electromagnetic field looks like as the field rotates with angular velocity w around z-axis. At first glance it would seem that we have a paradox: the linear velocity of motion well away from z-axis is getting more then speed of light. But motion of the field lines does not relate to transfer of matter or field energy. The above equations state only that the electromagnetic field at any point (r, 0, f) of space is equivalent to the value of field at the point (r, 0, f — ) at the moment t — Jf/w. Evidently,

in the far field zone only radiation field remains, which moves radially with the speed of light.

Substituting this function into Lagrange’s equation

(14)

d dL dL

dt dq dq

we obtain equations of motion for a charged particle in the field of rotating magnetic dipole

2

—^ [p — p02 — p(/ + w)2 sin2 0]

epw sin a

P

2

[0 sin / + - (/ + w) sin 20 cos /]

+

epw cos a

P

2

(/+ w)sin2 0 = 0, (15)

3 Equation of motion of a charged particle

Let us find the Lagrangian for a charged particle in the field under consideration [91:

A = -

(n x /) (n x /i)

mc2 1

—[2pp0 + p20-------p2(^/ + w)2 sin 20]

w2

+

2

euw sin a • 2

-------2-[/• sin / — 2/p sin2 0 cos /

(11) + Pw cos 20 cos /]

epw cos a

where v is the particle velocity vector, A is the vector potential. It is easy to prove that the vector potential

P

2

(/ + w) sin 20 = 0, (16)

—2-[p2/sin2 0 + p(/ + w)(p0sin20 + 2/Psin2 0)]

gives the fields (2) and (3). Substituting spherical com-Av

obtain

L

r2 + 0r2 + f 2r2 sin 0

1

+ — -j 0sin a(sin t + pcos t)

cr

+95 sin2 0 cos a

—f sin 0 cos 0 sin a(cos t — p sin t)} .

(12)

—c

"2W

pi2 + p202 + p2(/i + w)2 sin2 0

euw — sin a

c2 p

(/• + w) sin 0 cos 0 cos /

epw • . 2

+—^— cos a(/y + w) sin2 0.

c2 p

epw sin a .

+-------2----[sin 0 cos 0(pcos / — pw sin /)

+ 2p0 sin2 0 cos /]

Further we consider a charged particle to be a non-relativistic one. Let us introduce a new set of generalized coordinates p, 0, / with rw

p = —, / = f — wt. (13)

c

Actually, this means that we use a co-rotating reference system. As the particle is a non-relativistic one, we restrict our consideration by wr C c or p C 1. This means that the particle moving around the axis

w

non-relativistic. In this approximation t « —/.

Then the nonrelativistic Lagrangian function takes the form

—f 2

L

euw cos a , ■ „ 9 .

+-------2-(0psin20 — psin2 0) = 0. (17)

In order to find an integral of motion we multiply Eq. (15) by p, Eq. (16) by p0 and Eq. (17) by p/ and sum up all the three equations. The left-hand site of resulting equation is a full time derivative. After integration we obtain first order equation

—t(pi2 + p202 + p2/2 sin2 0 — w2p2 sin2 0)

2w2

epw2 sin 20 +----—------sin a cos /

2c2p

2

-----^2— cos a sin2 0 = E. (18)

c2 p

E

2

— c / 2 2 ' 2 2 2 "2\

K = —^(p + p 0 + p sin 0/ ) is always positive

2w2

hence, it can be considered as the kinetic energy of the particle. The rest terms are usually referred to as effective potential energy. It can be written as follows:

2

t r —C2 r 2-2/1

Vef = — {—p sin 0

N. sin 20 , 2N sin2 0]

+-------------cos /----------------- > , (19)

p

p

2

2

2

2

—c

2

c

where

N±

2 •

e^w2 sin a

C4 7

N

2

e^w2 cos a

C4 *

mc4 11 mc4

In this notation Eq. (18) can be represented as:

K = E - Vef *

(20)

0,

Excluding cot 0 from these equations we obtain:

2p6 - Nip3 - n2

0,

(27)

where N =

epw

. This equation has the solution:

p

N

T

; ± \J9 — sin2 a

p

N

T

cos a + q

\/9 — sin2 a

(28)

where q = e/|e| : for tan 0 we find:

±1 is the sign of the charge. And

tan 0 =

£ ^3 cos a + q9 — sin2 a)

2 sin a

(29)

Inequality E — Vef > 0 imposes restrictions on possible area of the particle motion.

4 Extremes of the effective potential energy

Let us investigate the effective potential energy. A particle can be in a stable, unstable or indifferent equilibrium at the stationary points of the effective potential energy. The aim of this investigation is to find the stationary points of the function Vef. In order to do so we have to solve the set of equations:

dVef

Hence, the solution of equations (22) - (24) for the sin 20 = 0

tive charge and two points for a negative charge.

nn

4.2 Solution for 0 = —

It follows from Eqs (22) - (24) that at the axis of rotation (0 = 0, n) all the first derivatives from effective potential energy are equal to zero only in the plane / = 0, n and for any p. Which means that on the axis

0 = 0, n there are not stationary points.

n

As to the equatorial plane 0 = —, Eqs (22) - (24) have the next solution:

(21)

p

cos 0 = 0, N > 0*

(30)

where qi = p, 0, 0. This gives a system of three equations

—2p3 + 2N|| - 2N± cot 0 cos 0 = 0, (22)

-p3 - 2N|| + 2N± cot 20 cos 0 = 0, (23)

N^ sin 20 sin 0 = 0* (24)

Equation (24) has next solutions:

0 = 0, n and any 0; (25)

nn

0 = —, (n G Z)^d any0* (26)

4.1 Solution for ÿ = 0, n

Using Eq. (25) we can eliminate the variable 0 from the equations (22) and (23) by substitution cos 0 = £, where £ = +1 corresponds to 0 = 0 and £ = -1 corresponds to 0 = n. This results in a system of two equations:

-p3 + N|| - £N^ cot 0 = 0,

-p3 cot 0 - 2Ni| cot 0 + £N^(cot2 0 - 1) = 0*

The sign before the square root is defined by the sign of the charge which is hidden in N, and condition p > 0. Therefore, the last equation takes the form

This gives two solutions for / = n/2 and / = 3n/2

According to inequality N- > 0 stated in (30) the a e cos a > 0

which means that the two above mentioned stationary points correspond to a positive charge if a < n/2 and to a negative charge if a > n/2.

Particle located at a stationary point can be in a state of equilibrium. Let us verify whether a particle with initial coordinates (pj,0j,/j) defined by Eqs (28) - (30) and initial zero velocity with respect to the rotating reference frame will be in equilibrium position. Substituting these coordinates and p = 0 = 0 = 0 in equations of motion (15) - (17) and taking into account that f = wt + / we obtain identical equalities. Hence, a particle being at rest at one of the stationary points in the co-rotating coordinate system will keep this position. This means that in laboratory reference frame the particle is moving along a circle with constant velocity vi = pic. Thus, there are at least six particular solutions of equations of motion which describe circuition of the particles in the field of inclined rotating dipole. Positions of the orbits defined by angle 0 and their radius are different for the positive and negative particle. Two of the trajectories are laying in z=0

5 Conclusions

We have recorded the components of the magnetic and electric fields of precessing magnetic dipole moment and equations of motion of a charged particle in this field. One first integral of motion was found. This made possible to introduce effective potential energy for the field of precessing magnetic dipole moment. All stationary points of the potential energy

4

mc

were found and it was shown that the stationary points Acknowledgement correspond to six particular solutions of equations of

motions. These solutions describe circular motion of a This research has been supported by the grant for particle with a constant velocity. LRSS, project No 224.2012.2.

References

[1] St0rmer C. Arch. Sci. Phys. Nat. 1907. V. 24. P. 5.

[2] St0rmer C. The polar aurora. Oxford: Clarendon Press 1955. 403 p.

[3] Alfven H. Cosmical Electrodynamics, International Series of Monographs on Physics. Oxford: Clarendon Press. 1950.

[4] Holmes-Siedle A. G., Adams L. Handbook of Radiation Effects. Oxford: Oxford University Press. 2002.

[5] DeVogelaere R. In: Contributions to the Theory of Nonlinear Oscillations, ed. Lefschetz S. Princeton: Princeton University

Press. 1958. P. 53-84.

[6] Dragt A. J. Rev. Geophys. 1965. V. 3(2). P. 255-298.

[7] Michel F. C. Theory of Neutron Star Magnetospheres. Chicago and London: The University of Chicago Press. 1991.

[8] Feynman R. P., Leighton R. B., Sands M. The Feynman Lectures on Physics. Vol. 2. NY: Addison-Wesley. 1964 .

[9] Landau L. D., Lifshitz E. M. The Classical Theory of Fields. 4th ed. NY: Pergamon. 1975.

Received 01.10.2012

В. Эпп, M. A. Macmepoea ПОЛЕ ПРЕЦЕССИРУЮЩЕГО МАГНИТНОГО ДИПОЛЯ

Исследовано поле прецессирующего магнитного диполя. Получен первый интеграл движения и найдены некоторые частные решения уравнений движения заряженной частицы в этом поле. Рассмотрена эффективная потенциальная энергия заряженной частицы. Показано, что эффективная потенциальная энергия имеет шесть стационарных точек, соответствующих движению заряда по окружности.

Ключевые слова: магнитный диполь, электромагнитное поле, наклонный ротатор, заряд, уравнения движения.

Эпп В. Я., доктор физико-математических наук, профессор.

Томский государственный педагогический университет.

Ул. Киевская, 60, Томск, Россия, 634061.

E-mail: epp@tspu.edu.ru

Мастерова М. А., аспирант.

Томский государственный педагогический университет.

Ул. Киевская, 60, Томск, Россия, 634061.

E-mail: Masterova@tspu.edu.ru