I have seen a website which is in Russian language and they expressed that the electric force line will bend along a axis if the charge keep moving, which will be looked like this: 
But could you please elaborate this reason with providing reasonable equation? When the speed is pretty close to the speed of light then the look will be much different. How will you explain this case?
E and B fields are elements of a second-rank tensor and hence their values in one inertial frame K' can be expressed in terms of the values in another inertial frame K. So, general Lorentz transformation from K to a system K' moving with velocity v relative to K must be in order. In true representation of these relations E and B fields are not independent of each other. A pure electrostatic field in one coordinate system wont transform into a purely magnetostatic field in another. Here is where it is more appropriate to talk about EM field rather than E or B separately. What you have here is an example of the transformation of fields seen by an observer in the system K when a point charge q moves by in a straight-line path with velocity v. The charge is at rest in system K' (attached to the charge), and the transformation of the fields is given by the relations (1,2) of the following post:
