4.4 Gravitational field

We have studied gravitational interaction in two related manners. First, we studied it in terms of force and then in terms of energy. There is yet another way to look at gravitational interactions. We can study it in terms of gravitational field.

In the simplest form, we define a gravitational field as a region in which gravitational force can be experienced. We should, however, be aware that the concept of force field has deeper meaning. Forces like gravitational force and electromagnetic force work with “action at a distance”. As bodies are not in contact, it is conceptualized that force is communicated to bodies through a force field, which operates on the entities brought in its region of influence.

Electromagnetic interaction, which also abides inverse square law like gravitational force, is completely described in terms of field concept. Theoretical conception of gravitational force field, however, is not complete yet. For this reason, we would restrict treatment of gravitational force field to the extent it is in agreement with well established known facts. In particular, we would not conceptualize about physical existence of gravitational field unless we refer “general relativity”.

A body experiences gravitational force in the presence of another mass. This fact can be thought to be the result of a process in which presence of a one mass modifies the characteristics of the region around itself. In other words, it creates a gravitational field around itself. When another mass enters the region of influence, it experiences gravitational force, which is given by Newton’s law of gravitation.

Field strength

Field strength ( E ) is equal to gravitational force experienced by unit mass in a gravitational field. Mathematically,

$$E=\frac{F}{m}$$

Its unit is N/kg. Field strength is a vector quantity and abides by the rules of vector algebra, including superposition principle. Hence, if there are number of bodies, then resultant or net gravitational field due to them at a given point is vector sum of individual fields,

$$E=E_1+E_2+E_3+\dots \dots .$$

$$\Rightarrow E=\Sigma E_i$$

Significance of field strength

An inspection of the expression of gravitational field reveals that its expression is exactly same as that of acceleration of a body of mass, “m”, acted upon by an external force, “ F ”. Clearly,

$$E=a=\frac{F}{m}$$

For this reason, gravitational field strength is dimensionally same as acceleration. Now, dropping vector notation for action in a particular direction of force,

$$\Rightarrow E=a=\frac{F}{m}$$

We can test this assertion. For example, Earth’s gravitational field strength can be obtained, by substituting for gravitational force between Earth of mass,"M", and a particle of mass, "m" :

$$\Rightarrow E=\frac{F}{m}=\frac{GMm}{r^{2}m}=\frac{GM}{r^2}=g$$

Thus, Earth’s gravitational field strength is equal to gravitational acceleration, “g”.

Field strength, apart from its interpretation for the action at a distance, is a convenient tool to map a region and thereby find the force on a body brought in the field. It is something like knowing “unit rate”. Suppose if we are selling pens and if we know its unit selling price, then it is easy to calculate price of any numbers of pens that we sale. We need not compute the unit selling price incorporating purchase cost, overheads, profit margins etc every time we make a sale.