Electromagnetic Repulsion Coil Calculations
The image shown is called a repulsion-ring demonstration. In the image, the iron rod passing through the ring helps to couple the magnetic field with the ring and prevent the ring from moving off to the side. Without the rod, repelling the conductive object will require a higher current and/or more turns in the coil. Also the direction of motion will be more difficult to control.

Alternating or pulsed current is required in the coil. The dimensions and specific material of the repelled object helps to determine the current induced and thus the force on the object.

The described technique is used in waste recycling to separate conductive material from non-magnetic waste materials. Magnetic waste is first separated by magnetic attraction.

Searching the internet for repulsion-coil information or information about magnetic separation of conductive materials may lead to specific information about designing such a system.

Others here may be able to offer more specific advice, but the question may be deemed too broad for this site.