This paper studies the mean first passage time (or exit time, or escape time) over the non-fluctuating potential barrier for a system driven only by a dichotomous noise. It finds that the dichotomous noise can make the particles escape over the potential barrier, in some circumstances; but in other circumstances, it can not. In the case that the particles escape over the potential barrier, a resonant activation phenomenon for the mean first passage time over the potential barrier is obtained.