The paper presents a theoretical investigation, using the perturbation technique, for the eddy currents induced in a conducting plate of finite thickness, due to two moving current filaments carrying direct current. The resultant magnetic field distribution, including the reaction effect of the eddy currents, is obtained in terms of first- and second-order effects of a parameter, which includes the velocity of the current filaments. The lift and drag forces on the current filaments, due to the eddy currents induced in the plate, are also investigated. The equations, formulated in terms of the vector potential, lead to the final results, which are expressed in non-dimensional form in order to appreciate the relative significance of the system dimensions and parameters. The results are original and novel, and provide an insight into the field phenomena associated with the eddy currents induced in a conducting plate which is in relative motion with a DC magnetic field. The analysis is of significance to systems involving repulsion support levitation and eddy current brake.