We developed an optical tweezers setup to study the electrophoretic motion of colloids in an external electric field. The setup is based on standard components for illumination and video detection. Our video based optical tracking of the colloid motion has a time resolution of 0.2 ms, resulting in a bandwidth of 2.5 kHz. This enables calibration of the optical tweezers by Brownian motion without applying a quadrant photodetector. We demonstrate that our system has a spatial resolution of 0.5 nm and a force sensitivity of 20 fN using a Fourier algorithm to detect periodic oscillations of the trapped colloid caused by an external ac field. The electrophoretic mobility and zeta potential of a single colloid can be extracted in aqueous solution avoiding screening effects common for usual bulk measurements.