The phenomena you tell about reminds me the description of galvanometer in a school grade book on physics.

Hello,

As we know that charge can obtained only on the spherical surface, so after putting sphere between the electrodes, which is acting somewhat capacitor or plates of opposite polarities, the sphere get attracted or repelled by the electrodes, means they generate magnetic field in presence of electric field. so we perfectly get the electromagnetic field, where you can use Maxwell's equations to find out related quantities.

Thanks,

A sphere halfway between two flat plates is equally attracted to both of them so it won't move.  The sphere will have electric charges on both sides, the opposite to the charges on the metal plate facing each side.  Things are never exact, so the sphere will be slightly closer to one of the electrodes and be attracted to that one.  When it touches the electrode, the sphere will conduct from the electrode and gain charge from the elctrode and be repelled by it, and move to the other plate.  This will continue.

https://www.youtube.com/watch?v=_iA5s6I05ak

Some time ago, several colleagues of mine studied the behavior of a conducting sphere inside a dielectric liquid (the conducting sphere was manipulated so as to have approximately the same mass density than the fluid, slightly heavier perhaps, in order to minimize gravity effects) and acted by the electrostatic field created by two flat electrodes. When the sphere touches the bottom electrode (say, negative electrode), it is charged with a negative amount of charge, thus being pushed towards the positive electrode. When the sphere touches the positive electrode, it transmits the negative charge to it and becomes positively charged. In this case the sphere is pushed towards the negative electrode and the process is reinitiated. The sphere is then a bouncing ball. Each time the sphere touches one of the electrodes, a peak of current can be registered.  The magnetic field does not play any role since the speed of the sphere is slow  and the involved characteristic times (charge and discharge processes) are very long. This is the simple framework. In the real situation the sphere was discharging through the liquid and before touching the flat electrodes the electric field between sphere and electrode is very high thus provoking interesting phenomena. I could get bibliography (a couple of papers) describing such work. If you are interested, please send me an email. 

The electric field and charge around and on a conducting sphere in a (previously) uniform electric field is a standard problem that is described in text-books.     e.g. see https://courses.cit.cornell.edu/ece303/Lectures/lecture8.pdf        If the sphere is much smaller than the width of the gap then this will be a good approximation to the correct field.  If the sphere is not much smaller than the gap then it becomes a more difficult problem.

Yes, Malcolm is right. The problem is that, independently of the size of the sphere, at some moment the distance between the sphere and one of the flat electrodes is not large in comparison with the radius of the sphere. Charge distribution on both the sphere and the flat electrode becomes more complicated. I think there is an analytical solution for this problem in the form of infinite series. The use of of the socalled bispherical coordinates would help. I suspect that an infinite series of images would also be a solution. In any case, not trivial at all. 

Many thanks for your kind answer for my question. These will help my studying a lot.