Gravitational potential and intensity

Question

Why don't we calculate gravitational field/intensity of a point due to a disc directly? We first calculate the potential then calculate the field by differentiating with respect to $dr$. Why don't we directly calculate the field?

Answer 1

When calculating the gravitational field of extended objects we typically break them up into infinitesimal masses then integrate to sum up the effect of all these masses. As it happens this process is described for a disk in the answers to the question The potential and the intensity of the gravitational field in the axis of a circular plate.

The advantage of working with the potential is that it is a scalar, so we can simply add up the potentials from all the infinitesimal masses to get the total potential. By contrast the field is a vector, so to sum up the field directly we would need to use vector addition to combine the fields from all the infinitesimal masses. This can be done, it just adds an extra level of complexity to the calculation. In most cases it is quicker and easier to work with the potential then differentiate it at the end.

Answer 2

Actually it is difficult to deal with vectors. Vectors have both magnitude and direction, and hence performing calculations involving vectors is a bit tedious(For example, in vector addition, we have to consider the directions of the different vector quantities and then add them using particular rules whereas scalar addition is elementary.) So people prefer to work with scalars and then convert the result to a vectorial one using the gradient operator.

In this case, Field intensity is a vector quantity whereas potential is a scalar quantity. Hence we prefer to calculate the scalar potential and compute the field intensity using grad operator.