Electron beam instabilites are studied by using a simple model for an electron beam streaming through a cold plasma, the beam being of finite width perpendicular to the ambient magnetic field. Through considerations of finite geometry and the coldness of the beam and background plasma, an instability similar to the two stream instability is assumed to be the means for wave growth in the system. Having found the maximum growth rate for one set of beam-plasma system parameters, this maximum growth rate is traced as these parameters are varied. The parameters that describe the system are the beam velocity (vb), electron gyrofrequency to ambient electron plasma frequency ratio (&OHgr;e/&ohgr;pe), the beam to background number density ratio (nb/na), and the beam width (a). When &OHgr;e/&ohgr;pe>1, a mode with &OHgr;e<&ohgr;<&ohgr;uhr is found to be unstable, where &OHgr; is the wave frequency and &ohgr;uhr is the upper hybrid resonance frequency. For low values of nb/na and &OHgr;e<&ohgr;pe, this mode is still present with &ohgr;pe<&ohgr;<&ohgr;uhr. If the beam density is large, nb/na?1, the instability occures for frequencies just above the electron gyrofrequency. This mode may well be that observed in laboratory plasma before the system undergoes the beam-plasma discharge. There is another instability present, which occurs for &ohgr;?&ohgr;pe. The growth rates for this mode, which are generally larger than those found for the &ohgr;?&ohgr;uhr mode, are only weakly dependent on &OHgr;d/&ohgr;pe. That this mode is not always observed in the laboratory implies that some factors not considered in the present theory suppress this mode, specifically, finite beam length. Consequently, this mode should be unstable in space plasmas, and it is possible that the &ohgr;?&ohgr;pe instability is of some significance for auroral zone electron beam instability. A more realistic electron distribution function should be employed to determine if a two-stream instability exists for the auroral plasma, but since the beam width is found to be important only for a<c/&ohgr;pe, finite geometry can be neglected.