In a uniform magnetic field, the Stern-Gerlach experiment would not result in the splitting of a particle beam. This is because the force acting on magnetic dipoles is dependent on the gradient of the magnetic field. Without a gradient, a uniform field exerts no net force on the dipoles. Mathematically, this can be expressed as the force being proportional to the field's gradient. Therefore, a uniform magnetic field fails to produce the observable effects seen in the experiment.