Journal of Electromagnetic Analysis and Applications, 2013, 5, 229-235
http://dx.doi.org/10.4236/jemaa.2013.55037 Published Online May 2013 (http://www.scirp.org/journal/jemaa)
229
Oscillations of a Punctual Charge in the Electric Field of a
Charged Ring: A Comparative Study
Najeeb Alam Khan, Asmat Ara, Nasir Uddin Khan, Nadeem Alam Khan
Department of Mathematical Sciences, University of Karachi, Karachi, Pakistan.
Email: najeeb@uok.edu.pk
Received July 25
th
, 2012; revised August 30
th
, 2012; accepted September 15
th
, 2012
Copyright © 2013 Najeeb Alam Khan et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We applied multiple parameters method (MPM) to obtain natural frequency of the nonlinear oscillator with rational
restoring force. A frequency analysis is carried out and the relationship between the angular frequency and the initial
amplitude is obtained in analytical/numerical form. This equation is analyzed in three cases: the relativistic harmonic
oscillator, a mass attached of a stretched elastic wire and oscillations of a punctual charge in the electric field of charged
ring. The three and four parameters solutions are obtained. The results obtained are compared with the numerical solu-
tion, showing good agreement.
Keywords: Oscillator; Stretched Wire; Parameters; Electric Field
1. Introduction
Nonlinear oscillators play a pivotal role in physics and
engineering. Recently, considerable attention has been
directed towards analytical/numerical solutions of nonlin-
ear equations. Many new techniques have been appeared
in writing, for example, max-min approach [1-3], frequency
amplitude formulation [4], homotopy methods [5-7], har-
monic balance method [8], parameter-expanding method
[9], variational approach [10-11], Hamiltonian methods
[12-14] and Lindstedt-Poincaré methods with modifica-
tion etc. [15-16].
In this paper, we consider a generalized nonlinear os-
cillator
2 1
2
0
1
n
m
u
u u
u
(1)
with initial conditions
0 u A and 0 0 u (2)
This equation occurs in certain phenomenon in rela-
tivistic physics, vibration of a stretched elastic wire due
to mass attached to the centre and oscillation of a punc-
tual charge in the electric field of charged ring. This
equation has been investigated by various authors [17,18]
for special cases.
It is interesting to note that 0, 1, 3 m reduce
to the oscillations of a charge in the electric field of a
charged ring equation. This connection is given as follows:
We consider a ring of radius r with a charge
spread uniformly around the ring. The electric field E on
the x-axis of the ring is given by
0 Q
32
2 2
1
4π
Qx
Ex
r x
(3)
where x is the distance along the axis. If a negative
punctual charge
Q Q
is placed at a point on the
ring axis, the charge will experience a force
32
2 2
1
4π
Q Qx
F x
r x
(4)
The equation of motion of the punctual charge is
given by the following nonlinear differential equation
Q
2
2 3
2 2
d 1
4π d
Q Qx
x
m
t
r x
2
(5)
Equation (5) can be written as
2
32
2 2 2
2
1d
d
x x
r x
r r t
(6)
where
3
4π
QQ
mr
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