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When the conductor is moved through the magnetic field, the charged particles within the conductor feel a magnetic force, equal to qvb. This magnetic force causes the electrons inside the atoms to "migrate" to one end of the rod. Having migrated to one end because of the force, that end is now negatively charged because it contains an excess of electrons. The other end of the conductor is now positively charged because the electrons have migrated away leaving a net positive charge.
The thing is that once we start having seperations of charge like this, we start to create electric fields. These electric fields that form because of the accumulation of charge cause and attraction between the postive and negative, essentially countering the magnetic force that was causing the initial seperation.
When enough charge has been accumulated in either end due to the magnetic force, the electric field will reach a point where it perfectly balances the magnetic force with its own electric one. This is equilibrium. The forces are now in balance and the charges will no longer increase.