Computationally efficient algorithms are presented for the computation of the effect of flat wire ends (end caps) in the common thin wire model. A uniform charge distribution over the surface of the end cap is assumed, and the full or exact kernel of the electric field integral equation formulation for cylindrical wires is used. The algorithms have been implemented in a highly efficient, low order, full kernel method of moments code for the analysis of relatively thick wire antennas and scatterers. The extra computational cost of including the end cap effect is small. The code has been applied to the analysis of a thick linear dipole and the results correspond very well with those of a recently published study using a much more computationally expensive implementation of the magnetic field integral equation with high order discretization methods.