Transformation of Surface Charge Density in Mixed Number Lorentz Transformation

Transcription

1 Sri ankan Journal of Phsics, Vol. 13(1) (1) 17-5 Institute of Phsics - Sri anka Research Article Transformation of Surface Charge Densit in Mied Number orent Transformation S. B. Rafiq * and M. S. Alam Department of Phsics, Shah Jalal Uniersit of Science & Technolog, Slhet-3114, Bangladesh. Abstract We know that the electric charge of an isolated sstem is relatiisticall inariant. We hae studied the transformation of surface charge densit in Special, Most general, Mied number orent transformation. As the formula of length contraction is not the same in these tpes of orent transformation, the transformation equation of surface charge densit will be different in the aboe mentioned transformations. Kewords: Special orent Transformation; Surface charge densit * 1. INTRODUCTION In most treatments on special relatiit 1, the line of motion is aligned with the - ais. In such a situation the (,) coordinates are inariant under the orent transformations. Howeer, it is of interest to stud the case when the line of motion does not coincide with an of the coordinate aes. Practical instances of such a situation are an airplane during landing or take off. The ground at the air field has a natural coordinate sstem with the -ais parallel to the ground, whereas the airplane ascends or descends at an angle with the ground. In this paper we derie the transformation equations for surface charge densit in different tpes of orent transformation when the line of motion is not aligned with an of the coordinate aes. * Corresponding Author E mail: sarwat7@gmail.com

2 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) Special orent transformation et us consider two inertial frames of reference S and S, where the frame S is at rest and the frame S moing along the X ais with elocit with respect to the S frame. The space and time coordinates of S and S are (,,, t) and (,,, t ) respectiel. The relation between then coordinates of S and S, which is called the special orent transformation, can be written as 1 t, (1a) =, =, (1b) (1c) t = t, (1d) where 1 and c = 1. 1 c And t (a) =, (b) =, (c) t t (d) 1. Most general orent transformation When the elocit of S with respect to S is not along the -ais, i.e. the elocit has three components, and, then the relation between the coordinates of S and S, which is called the Most General orent transformation, can be written as. / 1 t' t. Where, 1 and c = 1 1 c t The inerse most general orent transformation can be written as t t. / 1. t (3) 1.3 Mied-number orent transformation Mied-number orent transformation can be generated using Mied number algebra 5-9, 13. et us consider the elocit of the S frame with respect to the S frame is not along -ais i.e. the elocit has three components,,. et in this case and ' be the space part in S and S frame respectiel. In this case using equations (1a) and (1d), we can write

3 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) ( t' ) {( t ) ( t ) } ( t' ) {( t ) ( t )( )} (4) Using the product rule of two mied number algebra 6-8, we can write ( t )( ). t i Putting this in Eq. (4), we get ( t' ) {( t ) (. t i )} ( t' ) ( t. ) ( t i ) (5) The left hand side of equation (5) is the sum of a scalar t and a ector and the right hand side is also the sum of a scalar γ(t-.) [4] and a ector γ(-t-i ). So according to Mied number algebra, equating the scalar and ector parts of Eq. (5) we can write: ' ( t i ) (6a) t' ( t. ) (6b) Similarl it can be shown that ( t' i' ) (7a) And t ( t' '. ) (7b) Equations (6a), (6b), (7a) and (7b) are the Mied number orent transformation 5-7, 13.. SURFACE CHARGE DENSITY The surface charge densit is the amount of electric charge in a unit surface 3. In other words the amount of electric charge per unit surface area is called surface charge densit 3. It is measured in coulombs per square meter (C/m²). Since there are positie as well as negatie charges, the charge densit can take on negatie alues. ike an densit it can depend on position. We know that the charge on the electron or proton is the minimum, called the elementar charge e (= coul.). The electric charge is discrete which ma be determined b counting the number of elementar charged particles. As the total number of elementar charges cannot depend on the state of the motion of the obserer, we ma conclude that the electric charge is relatiisticall inariant. Based on this important conclusion we want to calculate the transformation equation for the charge densit σ.

4 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) TRANSFORMATION OF SURFACE CHARGE DENSITY 3.1 Transformation of Surface Charge Densit using Special orent transformation et us consider two sstems S and S', S' is moing with uniform elocit relatie to S along negatie direction of X-ais as shown in Fig. 1. et there be a stationar sheet of uniform charge densit +σ coul/m at rest in sstem S haing one edge parallel to X'-ais. et the sheet be a square of side and placed parallel to X-Y plane. The obserer in the sstem S' will obsere that the sheet is moing with elocit along (+) e X-ais. Z Figure 1: The sstem S' is moing along X-ais with elocit relatie to the sstem S. We hae the transformation relation of surface charge densit from the principle of inariance of charge which states, The total electric charge in an isolated sstem is relatiisticall inariant. The total charge as obsered in the sstem S is Q (8) The obserer in sstem S' will noticed that the side of the square along X-ais has been contracted from to, where 1 c As the length is contracted onl along the X-ais, the charge as obsered b obserer in the sstem S' is Q 1 c Where σ' is the surface charge densit in sstem S'. According to the principle of conseration of charge Q Q (9)

5 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) c [Using Eqs. (8) and (9)] 1 c This equation represents the transformation equation for the surface charge densit 3 in special orent transformation (Table 1). 3. Transformation of surface Charge Densit using Most General orent transformation et us consider two sstems S and S' where the frame S' is moing with elocit relatie to the sstem S along X-Y plane as shown in Fig.. Thus the elocit of has two components and. et us consider a stationar sheet of uniform charge densit +σ coul/m at rest in sstem S haing one edge parallel to X-ais and let the sheet be a square of side placed parallel to X-Y plane. The obserer in the sstem S' will obsere that the sheet is moing in opposite direction. S' S Figure : The sstem S' is moing in X-Y plane with elocit relatie to the sstem S. If is the length of the square charged sheet in S, then the length contraction in the moing frame for the most general orent transformation can be written as 1 ' cos 1' cos 1 cos 1 cos 1. cos 1 = 1 cos 1 cos 1 1 cos 1 cos 1

6 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) cos 1 1 cos 1 cos 1 cos 1 and 1 cos 1 1 cos The total charge obsered b an obserer in S' is Q 1 cos. 1 cos 1 1 cos 1 cos 1 ' 1 cos ' [for square charge sheet] (1) 1 cos 1 Where σ' is the surface charge densit in sstem S'. According to the principle of conseration of charge Q Q Using Eqs. (8) and (1), this can be written as 1 cos 1 cos 1 1 cos 1 cos 1 This equation represents the transformation equation for surface charge densit when the sstem S' is moing in X-Y plane (Table 1). 3.3 Transformation of Surface Charge Densit with Mied number orent transformation et us consider two sstems S and S' where the sstem S' is moing with the elocit relatie to the sstem S along an arbitrar direction as shown in the figure- so that the elocit of hae two components and. et us consider a stationar sheet of uniform charge densit +σ coul/m at rest in sstem S haing one edge parallel to X-ais and let the sheet be a square of side placed parallel to X-Y plane. The obserer in the sstem S' will obsere that the sheet is moing in opposite direction with elocit along an arbitrar direction as shown in the figure.

7 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) If is the length of the square charged sheet in S, then the length contraction in the moing frame for the mied number orent transformation can be written as 1 ( i ) [. i( ) i( ). i( ). i( ) [ i ( cos )] = [1 (1 cos )] [1 (1 cos )] [1 (1 cos )] [1 (1 cos )] and [1 (1 cos )] The total charge as obsered b an obserer in S' is Q [1. (1 cos )] [1 (1 cos )] [1 [for square charge sheet] (11) (1 cos )] Where σ' is the surface charge densit in sstem S'. According to the principle of conseration of charge Q Q [1 (1 cos )] [1 (1 cos )] [using Eqs. (8) and (11)] This equation represents the transformation equation for the surface charge densit in Mied number orent transformation (Table1).

8 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) Table 1: Transformation equations for surface charge densit in different tpes of orent transformation (T). Space Special T Most general T Mied-number T ' t, 1 1 / 1 / ' t i ( ) 1 ' t / t i ( ) =, ' t i ( ) 1 / 1 1 / =, 1 / t Time t ( t ) 1 / 1 1 / 1 t / t t t' ( t ) ength contraction Surface charge densit cos 1 i [1 cos ( 1) cos ( 1) [1 (1 cos )] Table : Numerical alues of surface charge densities of moing sstem in terms of that of rest sstem in Different tpes of orent transformation (T) Parameters Special T Most general T Miednumber T θ = =.5c σ - - =.7c 1.4 σ - - θ = 3 =.5c σ 1.5 σ =.7c σ 1.73 σ θ = 45 =.5c σ 1.16 σ =.7c σ 1.48 σ θ = 6 =.5c σ 1.8 σ =.7c σ 1.4 σ * calculations were carried out taking c as unit.

9 S. B. Rafiq et al. /Sri ankan Journal of Phsics, Vol. 13(1), (1) CONCUSION The transformation formula for surface charge densit in Special, Most general, Mied number orent transformation are illustrated in Table 1. It has been obsered that the formula of the surface charge densit in case of most general orent transformation is more complicated than mied number orent transformation. The numerical alues of surface charge densities of the moing sstem in terms of a sstem at rest for different tpes of orent transformation were calculated as shown in Table. From Table it is obsered that for the same angle (θ) between the sstems S and S, the alue of surface charge densit of moing sstem (σ ), increases with increasing elocit of the moing sstem and for the same elocit of the moing sstem σ decreases with increasing the angle θ. Although the results obtained in case of most general orent transformation and mied number orent transformation are the same, the calculation is easier in case of mied number orent transformation. So, it can be conenient to use mied number orent transformation when the line of motion does not coincide with an of the coordinate aes. REFERENCES 1. R. Resnick, Introduction to Special Relatiit (Wile Eastern limited,1994).. Moller, The Theor of Relatiit (Oford Uniersit Press, ondon, 197). 3. S. Prakash, Relatiistic Mechanics (New Delhi: Progati Prakashan, ). 4. M. R. Spiegel, Theor and Problems of Vector Analsis and An Introduction to Tensor Analsis (Schaum s outline series, McGraw-Hill book compan). 5. M.S. Alam, Mied Product of ectors, Journal of Theoretics. 3, (1) M.S. Alam, M.H. Ahsan and M. Ahmad, Mathematical Tools of Mied-Number algebra, J. Natn. Sci. Foundation, Sri anka. 33, (5) M.S. Alam, M.H. Ahsan, Mied-Number orent Transformation, Phsics Essas, 16, (3) M.S. Alam, Consequences of Mied Number orent Transformation, Proc. Pakistan Acad. Sci. 41, (4). 9. M.S. Alam, M.D. Chowdhur, Relatiistic aberration of Mied number orent transformation, J. Natn. Sci. Foundation Sri anka. 34, (5) M.S. Alam, Comparatie Stud of Mied Product and Quaternion Product, Indian Journal of Phsics A 77, (3). 11. M.S. Alam, Comparatie Stud of Quaternions and Mied Number, Journal of Theoretics. 3, (1) M.S. Alam, K. Begum, Different tpes of orent transformation, Jahangirnagar Phsics Studies. 15, (9). 13. M.S. Alam, S. Bauk and M. Ahmed, Agreements and disagreements between orent transformation and on Peschke s comment, Phsics Essas 4, (11) 1.