yes. the lower you go the more you capture, but you usually lose some detail on the edges. just like with my 10-20mmi can capture both walls of my room and lets say with my 50mm I can't
Just outta curiosity...are you meaning "wide" as in physical width of the lens, or description as wide angle?
If you mean physical width...the width has more to do with aperture size than field of view. The larger aperture allows more light in due to the increased size of the aperture, over a smaller (narrower) lens.
Most wideangle lenses are better recognized by there focal #. IE...28mm, 35mm, etc...usually the lower the number the wider the view.