Recently the Casimir effect has been getting more popular because of

its importance in designing micro and nano scale machines. Working in

the language of quantum field theory simplified formulas for the

Casimir energy for a massless scalar field are worked out. These

formulas are applied to planar potentials and potentials in the

annular region between two co-axial cylinders. A scalar equivalent to

the Lifshitz formula is applied to new cases of non-trivial planar

potentials, specifically two interacting linear potentials and two

interacting quadratic potentials. In addition many exact expressions

for the Casimir energy between two weakly coupled objects are worked

out for many non-trivial geometries. Exact closed form results are shown

for parallel cylinders, spheres, and finite ribbons and plates. These

closed form results are used to check the range of validity of the

proximity force approximation.