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    Relativistic Magnetic Field

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    bhooshan jape
    This document presents a simple relativistic derivation of the electric and magnetic fields generated by an electric point charge moving with constant velocity. The derivation is based on using radar detection to determine the point in space and time where the fields are measured from the reference frame of the observer. This approach provides a more straightforward derivation than previous methods that relied solely on general electromagnetic field equations without using retarded potentials or relativistic transformations. The derivation recovers the classic formulas for the electric and magnetic fields generated by a moving point charge.

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    Relativistic Magnetic Field

    Uploaded by

    bhooshan jape
    153 views5 pages
    This document presents a simple relativistic derivation of the electric and magnetic fields generated by an electric point charge moving with constant velocity. The derivation is based on using radar detection to determine the point in space and time where the fields are measured from the reference frame of the observer. This approach provides a more straightforward derivation than previous methods that relied solely on general electromagnetic field equations without using retarded potentials or relativistic transformations. The derivation recovers the classic formulas for the electric and magnetic fields generated by a moving point charge.

    Original Description:

    learn about the relatisvistic nature of magnetic field. detailed mathematical analysis is provided

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    This document presents a simple relativistic derivation of the electric and magnetic fields generated by an electric point charge moving with constant velocity. The derivation is based on using radar detection to determine the point in space and time where the fields are measured from the reference frame of the observer. This approach provides a more straightforward derivation than previous methods that relied solely on general electromagnetic field equations without using retarded potentials or relativistic transformations. The derivation recovers the classic formulas for the electric and magnetic fields generated by a moving point charge.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    153 views5 pages

    Relativistic Magnetic Field

    Uploaded by

    bhooshan jape
    This document presents a simple relativistic derivation of the electric and magnetic fields generated by an electric point charge moving with constant velocity. The derivation is based on using radar detection to determine the point in space and time where the fields are measured from the reference frame of the observer. This approach provides a more straightforward derivation than previous methods that relied solely on general electromagnetic field equations without using retarded potentials or relativistic transformations. The derivation recovers the classic formulas for the electric and magnetic fields generated by a moving point charge.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    Relativistic derivations of the electric and magnetic fields generated by

    an electric point charge moving with constant velocity


    Bernhard Rothenstein1), Stefan Popescu 2) and George J. Spix 3)
    1) Politehnica University of Timisoara, Physics Department, Timisoara, Romania,
    bernhard_rothenstein@yahoo.com
    2) Siemens AG, Erlangen, Germany, stefan.popescu@siemens.com
    3) BSEE Illinois Institute of Technology, USA, gjspix@msn.com

    Abstract. We propose a simple relativistic derivation of the electric and the magnetic
    fields generated by an electric point charge moving with constant velocity. Our approach
    is based on the radar detection of the point space coordinates where the fields are
    measured. The same equations were previously derived in a relatively complicated way2
    based exclusively on general electromagnetic field equations and without making use of
    retarded potentials or relativistic equations.

    1. Introduction
    Heaviside1 obtained the expressions for the electric and magnetic
    fields produced by an electric point charge q moving with constant velocity
    V given by
    E=

    q
    1% " 2
    4#$ 0 r 3 1 % " 2 sin 2 !

    3/ 2

    (1)

    (2)
    where r is the distance between the point of observation and the charge, c is
    the velocity of light and ! is the angle between r and V.
    Jefimenko2 presents the ways in which (1) and (2) could be derived.
    Among the four different approaches that he mentions, one particular
    approach is based on applying Lorentz-Einstein transformations of spacetime coordinates and fields to a stationary point charge.3 He considers that
    this method is fairly simple mathematically, but it doesnt fit into the
    classical theory of electromagnetism.
    In the literature covering this subject we found the following formulas
    B = c "2 V ! E

    q
    1 ' . 2 & rV #
    r'
    c !"
    4/0 0 r 3 - r.v * 3 $%
    +1 '
    (
    rc )
    ,
    q
    1( ) 2
    [V ! r ].
    B = 0
    4*r 3 ' r.V $ 3
    %&1 ( rc "#

    E=

    (3)

    (4)

    Jefimenko2 derives them based exclusively on general electromagnetic


    field equations and do not make use of retarded potentials or relativistic

    equations. These derivations are relatively complicated making hard theirs


    teaching without mnemonic helps.
    The purpose of our paper is to present the relativistic derivations of
    (1) and (2) and of (3) and (4) in a simple and transparent way showing the
    difference between the physics behind them. We start with the hints that the
    relativists always have in mind:
    When you speak about a physical quantity then always define it
    without ambiguity mentioning the observer who measures it, the measuring
    devices he uses and where and when he performs this measurement.
    2. The electric and the magnetic fields of an electric point charge
    moving with constant velocity.
    The scenario we propose involves the inertial reference frames
    K(XOY) and K'(X'O'Y') with parallel axes, K' moving with constant speed V
    relative to K in the positive direction of the common OX(O'X') axes. At a
    time t=t'=0 the origins O and O' coincide in space. A point charge q is at
    rest in K' being located at its origin O' as shown in Figure 1. The electric
    field generated by this charge (stationary in K') is measured at a point
    M "( x " = r " cos ! ", y " = r " sin ! ") by the observer R "( x " = r " cos ! ", y " = r " sin ! ") located
    at this point. Assuming that the Coulombs law holds true in the rest frame
    of the charge, then the point charge q creates a radial electric field in K'
    E! =

    q
    kq
    = 2
    2
    4"# 0 r !
    r!

    (5)

    The components of which are


    E x" =

    kq
    cos ! "
    r "2

    (6)

    E "y =

    kq
    sin ! " .
    r "2

    (7)

    and
    In its rest frame K' the charge doesnt generate a magnetic field thus B'=0.
    Y'

    E'
    M'

    y'
    q
    O'

    r'

    "'
    x'

    X'

    Figure 1. Scenario for deriving the electric and magnetic fields generated
    !y
    by a uniformly moving point charge inEits
    rest frame K' (X'O'Y').
    E x!

    The first problem we have to solve is to find out a relationship


    between the field E generated by the charge as measured by observers from
    K and the same field E' measured by observers from K'. Thought
    experiments involving infinite extending charged surfaces or lines lead to4
    E x = E x!
    (8)
    and
    Ey =

    E (y
    &V #
    1' $ !
    %c"

    (9)

    (6) and (7) becoming


    kqx (

    E x( =
    r(

    Ey =
    r (3

    (10)

    &V #
    1' $ 2 !
    %c "
    kqy (
    &V #
    1' $ !
    %c"

    (11)

    It is customary in special relativity to express the right sides of (10)


    and (11) as a function of physical quantities measured in K. Because the
    charge is a velocity independent physical quantity, a fact best proved by the
    neutrality of an atom inside which negative electrons move with different
    values around the positive nucleus, the only thing we have to do is to express
    x !, y !, r ! as a function of x, y, r via the Lorentz-Einstein transformations.
    There are two possible approaches. In the first one, observers from K
    detect simultaneously the two ends of the position vector of point M', a
    procedure associated with the two simultaneous events E0 (0,0,0) and
    E ( x, y,0) resulting that
    x( =

    r cos )
    &V #
    1' $ !
    %c"

    (12)

    (13)

    y " = y = r sin !
    r( = r

    1 ' * 2 sin 2 )
    &V #
    1' $ !
    %c"

    (14)

    Equations (10) and (11) become


    E x = kq

    (1 # " 2 ) cos !

    r 2 1 # " 2 sin 2 !

    3/ 2

    (15)

    E y = kq

    (1 " ! 2 )

    or
    E = k (1 # " 2 )

    r
    3

    r 1 # " sin 2 !

    3/ 2

    (16)

    recovering (1).
    In a second approach, which is more in the spirit of classical
    electromagnetism, we consider that the information about the fact that
    charge q arrives at t'=0 at point O'(0,0) propagates in free space with speed c
    and arrives at a point M "( x " = r " cos ! ", y " = r " sin ! ") at a time t ! =

    r!
    generating
    c

    r!
    c
    start of the information from O' is E0! (0,0,0) . The same events detected from
    r
    K are E0 (0,0,0) and E ( x = r cos ! , y = r sin ! , t = ) . Appling the Lorentz-Einstein
    c

    the event E !( x ! = r ! cos " !, y ! = r ! sin " !, t ! = ) . The event associated with the

    transformations we obtain
    x( = r

    cos * ' )
    &V #
    1' $ !
    %c"

    (17)

    (18)

    y " = y = r sin !
    1 ' * cos )
    r( = r
    2
    &V #
    1' $ !
    %c"

    (19)

    which with (10) and (11) become in this case


    E x = kq

    (cos ! # " )(1 # " 2 )


    3
    r 2 (1 # " cos ! )

    (1 # " )sin !
    2

    E y = kq

    r 2 (1 # " cos ! )

    (20)
    (21)

    or
    E=

    kq
    1' ) 2
    V#
    &
    r'r !.
    3
    3 $
    c"
    r (1 ' ) cos ( ) %

    (22)

    By this simple deduction we recover Jefimenkos result2. However his


    original deduction appears to us as a veritable tour-de-force involving a
    great number of intermediary steps.
    4

    The magnetic field can be calculated using the well known formula
    that relates the electric and the magnetic field
    (23)
    B = c "2 V ! E .
    We consider that our second approach to derive the space coordinates
    of the point where the field is measured involving a radar detection
    procedure is more in the spirit of electrodynamics than the first approach
    involving the simultaneous detection of the moving rod coordinates5. We
    also point out that the radar detection shares much in common with the
    concept of retardation.
    Conclusions
    We have presented a new method to calculate the electric and
    magnetic fields generated by an electric charge moving with constant
    velocity. Our method is based on the radar detection of the point space
    coordinates where the fields are measured. This calculation is much simpler
    than an alternative approach2 based exclusively on general electromagnetic
    equations which doesnt make use of retarded potentials or relativistic
    equations. In accordance with a hint of Ockham we propose the use of our
    simpler deduction in teaching relativistic electrodynamics as it is more
    transparent and time saving.
    References
    Oliver Heaviside, The electromagnetic effect of a moving charge, The
    Electrician, 22, 147-148 (1888)
    2
    Oleg D. Jefimenko, Direct calculation of the electric and magnetic fields
    of an electric point charge moving with constant velocity, Am.J.Phys. 62,
    79-85 (1994)
    3
    W.G.V.Rosser, Classical Electromagnetism via Relativity, (Plenum, New
    York, 1968) pp. 29-42
    4
    Edward M. Purcell, Berkeley Physics Course Volume 2 (McGraw-Hill,
    1960)
    5
    W.G.V. Rosser, Comment on Retardation and relativity; The case of a
    moving line charge, by O.D. Jefimenko (Am.J.Phys. 62, 79-85 (1994))
    Am.J.Phys 63, 454-455 (1995)
    1

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