Volome Charge Density in Mixed Number Lorentz Transformation

We know charge density is changed when it observes from a moving frame of reference due to the length contraction. In this paper we have studied the volume charge density in special and mixed number Lorentz transformation. We also investigate the changes of the volume charge density of moving system in terms of rest system in mixed number Lorentz Transformations at different angles and velocities.


Introduction
Lorentz Transformation is the relation between two coordinate frames where one is in uniform motion with respect to the other. In special relativity the line of motion is placed in the x-axis. Here the interest is to observe the situation when the line of motion does not coexist with any of the coordinate axes. Charge density means how much charge is present in a given length, area or volume. Therefore, volume charge density tells specifically about how much electric charge is present in a given volume [9]. Rafiq and Alam [1] have derived the transformation equation of surface charge density in mixed number Lorentz Transformation. Bhuiyan et al. [2] have derived the transformation equation of surface charge density in different types of Lorentz transformation. In this case a cube is used to find out the formula of volume charge density in mixed number Lorentz transformation. In Lorentz transformation length is contracted, so the volume charge density will be different in different types of Lorentz transformations. In this paper the transformation equation for volume charge density in mixed number Lorentz transformation has been derived.

Special Lorentz Transformation
Let and be two frames of references, where is at rest and is moving along x-axis with velocity with respect to frame.The space and time coordinates of and are and respectively. The relation between the coordinates of and which is called the special Lorentz Transformation [1][2][3]9], can be written as where, √ and The inverse Lorentz Transformations [1][2][3]9] can be written as (5)  (6)  (7)  (8) where, √ and

Mixed Number Lorentz Transformation
Mixed number is the extension of complex number [10]. Mixed number Lorentz transformation can be formed using mixed number algebra. Let us consider the velocity of frame with respect to frame is not along the -axis i.e the velocity has three components , . Consider in this case and be the space part in and . Then we can write [1][2][3]9] (9) (10) Similarly we can write (11) (12) Eqs. (9), (10), (11) and (12) are the mixed number Lorentz transformation.

Volume Charge Density
Volume charge density is the amount of electric charge per unit volume. It is denoted by .
It is measured by coulombs per cubic meter ⁄ . We know that the charge on the electron and proton is the minimum, called the elementary charge e The electric charge is distinct which may be determined by counting the number of elementary charged particles. It is known that the total number of elementary charges cannot depend on the state of the motion of the observer therefore we may conclude that the electric charge is relativistically invariant. Based on this conclusion we have derived the transformation equation for the volume charge density [4][5][6]9].

Transformation of Volume Charge Density in Mixed Number Lorentz Transformation
Let two inertial systems and where the frame is moving with uniform velocity relative to the rest system in arbitrary direction as shown in fig.1. Thus velocity has three components , , . Let us consider a stationary cube of uniform charge density ⁄ at rest in system having one edge parallel to -axis and let the length of the side of the cube is . The observer in system will observe that the cube is moving with velocity which is parallel to -axis. Fig. 1. The system is moving with velocity relative to the system If is the length of the cube in , then the length contraction in the moving frame for the mixed number Lorentz transformation can be written as [8] , and The total charge as observed by an observer in is  Table 1 Table1. Transformation equation for VCD in MNLT.
Length contraction Volume charge density

Conclusion
The transformation formula for volume charge density in Mixed number Lorentz transformation is illustrated in Table 1. The numerical values of volume charge densities of the moving system in terms of a system at rest for mixed number Lorentz transformation have been calculated and it is observed graphically that for same angle between the system and , the values of volume charge density of moving system increases with increasing velocity of the moving system. On the other hand for the same velocity of the moving system the values of volume charge density decreases with increasing the angle .