$$\require{cancel}$$

# 10.1.5: Superposition

• • Wendell Potter and David Webb et al.
• Physics at UC Davis

So far the general principles of fields have been introduced by using a spherical or point mass as an example. For this case, we have a general formula for the gravitational field $\mathbf{g}_{M} = \dfrac{GM}{r^2}$ where the direction is always pointed toward the mass $$M$$.

If we have two point-like or spherical masses, then we use superposition. When we used this word with waves, this meant that we added the two waves together to find the total displacement. When we use it for fields this means that we figure out what the field from the first source is, then figure out the field from the second source separately, and add them together. It is critical to remember in the last step that we are adding them as vectors (a review of how to add vectors is provided here). Another important point to remember: we will never add field vectors at different locations. Any time we “superpose” two or more different sources we will be looking at their effect at the same location at the same time. It's worth noting that superposition of gravitational fields (in the order of magnitude that we work) is an experimental conclusion, not a conclusion we can reach with math alone.

Example #3

Draw a vector map for the gravitational field of two separated spherical balls of equal mass. I.e. pick a reasonable number of points at which to evaluate the field, and draw a rough field map.

Solution

Let us start by drawing two pictures of the situation. On the picture on the left we only include the field from “ball 1”, and on the right we only include the field from “ball 2”. These sketches are only rough, but they show that as we get further from the source the field gets weaker, and the direction of the field is always toward the source.  Now we have to put these two fields together, and add the fields as vectors. On the left hand side we display both of the fields before doing the vector sum, and on the right is the total vector field (i.e. after doing the vector sum).  We now have enough information to deal with an arbitrary distribution of mass. Here's how; we can think of taking any distribution of mass and breaking it into a lot of different point masses. We can figure out the field from each of these point masses on any location, and then add the contributions from each point together to find the total gravitational field. This would involve a lot of work (it can be simplified using integral calculus), but at least we know how to do it in principle.