Anyone here have an equation to predict the forces given by an electrotromagnet on an object based on wire size current and coil diameter? I want to replace a mechanical lever system and think this maybe a useful equation for a feasibility study if I can get some understanding how these magnet systems work. Any help most appreciated in advance.

I'm afraid the equation is anything but straightforward. I just did this quest recently. Here's a few rules of thumb to get you started... The optimal outer diameter of a coil of wire should be 3 times the inner diameter. The optimal wire diameter is directly proportional to the coil diameter. The length of wire should be enough to provide about the same electrical resistance as your power source.

I want to second this rec. FEMM is open source. I have never seen anything other program that is cost free. It is s rather clunky interface - 2D only, but it is worthwhile to try this out. It takes very little complexity before hand calculations become unmanageable; FEMM comes in handy when the complications are getting difficult. Once you have created the wire frame (importing DXF works), it is relatively easy to try out different materials and properties, change dimensions, and display the results in different ways (force and equipotential are available). This is a great way to get a an intuitive grip on how a magnetic circuit is behaving.