How do you calculate the expected force of an electromagnet?

I agree that this question may be a duplicate, but previous questions seemed to be focused on particular geometries and situations. Here's my two cents..

The Lorentz force is

$J times B$

You might combine Maxwell's equations (while neglecting the timescales for charges to move to conductor surfaces ($frac{partial mathbf{E}}{partial t}$)) to get the induction equation

$frac{partial mathbf{B}}{partial t} + nabla times left(sigma^{-1} left[ nabla times frac{mathbf{B}}{mu} right] right) = nabla times (mathbf{u} times mathbf{B}), qquad nabla bullet mathbf{B} = 0.$

Assuming there's no moving conductor, we may write this as

$frac{partial mathbf{B}}{partial t} = - nabla times left(sigma^{-1} left[ nabla times frac{mathbf{B}}{mu} + mathbf{J}_{external} right] right).$

Where $mathbf{J}_{external}$ is the externally applied current. The magnetic field can be solved from the induction equation, and current ($mathbf{J}$) may be found from ampere's law $mathbf{J} = nabla times frac{mathbf{B}}{mu}$

In summary, the induction equation can help give you some intuition as to how the magnetic field behaves, this gives you information about the magnetic field and current (via ampere's law).

Hope this helps.