The task of an umbrella light is to convert a petite light source so that it functions as a broad light source. Small light sources function almost as if they are point light sources. In other words, the output is much like the output of a bare light bulb. Such lamps cast harsh shadows. Also, a point light source obeys precisely the law of the inverse square.
When we talk about the a lamp obeying the law of the inverse square, we are talking about the way the light falls off with distance. The essence of this law is: Double the distance, lamp to subject and the fall off is 4X. Stated differently, if a lamp is 4 feet from the subject, and you increase this distance to 8 feet, the light level falls to only 25% of the original brilliance. That’s 2 f-stops reduction. Such an enormous change in exposure will be disastrous if the subject is moving about on the set. If instead of a point light source, a broad is used, the law of the inverse square is defeated. That’s why broad light sources are commonly used on a movie sound stage. The use of a broad maintains a relativity constant exposure over a wide distance range.
Besides the fact that a broad lamp defeats the law of the inverse square, a broad produces a highly diffused light. Such a light casts no harsh shadows, and thus is favored when soft light enhances the subject. Umbrellas are broad lamps -- and bigger is better.