Brownian motion conned in a two dimensional channel with varying crosssectionunder the inuence of an external force eld is examined. In particular,a one dimensional equation approximately describing the dynamics of the Brownianparticles is derived, a generalization of the well known Fick-Jacobs equation.This generalized Fick-Jacobs equation is numerically veried by Browniandynamics simulations for a special case of the external force eld. Furthermorethe generalized Fick-Jacobs equation is investigated in the context of stochasticthermodynamics.