Definition of Gravitational Field:
The region around an object in which another object experiences a gravitational force then the region of that object is called the gravitational field.
Gravitational Field Intensity:
The force applied per unit mass of an object that is placed in the gravitational field is called the intensity of the gravitational field.
$\overrightarrow{E}=- \frac{G \: M}{r^{2}} \hat{r}$
The Expression for gravitational field intensity:
Let us consider
The mass of a lighter object that experience the force = $m$
The mass of a heavy object that produces the gravitational field= $M$
The distance between the objects = $r$
The gravitational force between the objects is
$F=G\frac{M\:m}{r^{2}} \qquad(1)$
Now the force per unit mass i.e Gravitational field intensity
$E=-\frac{F}{m} \qquad (2)$
Here the negative indicates that the direction of force is opposite to $\hat{r}$
The vector form of the gravitational field intensity
$\overrightarrow{E}=-\frac{F}{m} \hat{r}$
Where $\hat{r} \left (=\frac{\overrightarrow{r}}{r} \right)$ is the unit vector along the $\overrightarrow{r}$
Now substitute the value of $F$ in the above equation $(2)$. Therefore we get,
$E=-G \frac{M\:m}{m r^{2}}$
$E=- \frac{G \: M}{r^{2}}$
The vector form of the above equation
$\overrightarrow{E}=- \frac{G \: M}{r^{2}} \hat{r}$
