The Limiting Magnitude of a 12" clear aperture telescope is 2.7 + (5 * log D) where D is the mirror's full aperture in millimeters, which is (14 * 25.4) = 355.6 mm giving us a limiting magnitude of
2.7 + (5 * log 355.6) = 15.45  Objects dimmer than this magnitude will not be visible.  Dimmer objects will have a higher Apparent Magnitude.
An object's Apparent Magnitude depends on its size, distance, and brilliance. The latter depends on how bright the object burns if it is a star, or on its reflecting ability called albedo if it isn't a star. This should not be confused with its Absolute Magnitude which is the change in Apparent Magnitude that would result from moving the object inwards or outwards to a standard distance of 10 Parsecs away if it is a star, or to a standard distance of 1 Astronomical Unit, AU, if it is a planet, asteroid, or comet.
Over the years Astronomers have developed a family of equations that solve for various combinations of the given variables and a telescope's aperture so that the Absolute Magnitudes, Apparent Magnitudes, and distances can be calculated from known diameters and albedos.
A 20 km asteroid located 0.03939411 AU beyond Earth with an albedo of 0.1 will have an Absolute Visual Magnitude of:
V$$_abs = V$$_Sun - 2.5 * log$$₁₀((Diam$$_Obj / 2)$$² * A) = 14.1 - 2.5 * log$$₁₀((20/2)$$² * 0.1) = 14.1 - 2.5 * 1 = 11.6 If it is seen in full phase, it will have an Apparent Visual Magnitude of:
V$$_app = V$$_abs + 5 * log$$₁₀(d * D + φ) = 11.6 + 5 * log$$₁₀(0.03939411 * 1.03939411 + 0) = 11.6 + (- 6.94) = 4.67 where:
d is the distance between the astroid and Earth in AU
D isthe distance between the astroid and the Sun in AU
φ  is the phase of the astroid as viewed from Earth