4.5 Relation between gravitational field and gravitational potential

1.1 Characteristics of gravitational force
1.2 Universal gravitational constant
1.3 Principle of superposition of gravitation
1.4 Gravity
1.5 Acceleration due to gravity
1.6 Relation between $g$ and $G$

2.1 Variation of acceleration due to gravity $(g)$ due to shape of the earth
2.2 Variation of acceleration due to gravity $(g)$ due to altitude
2.3 Variation of acceleration due to gravity $(g)$ due to depth
2.4 Variation of acceleration due to gravity $(g)$ due to rotation of earth

3.1 Gravitational field due to a point mass
3.2 Gravitational field due to a uniform solid sphere
3.3 Gravitational field due to a uniform spherical shell
3.4 Gravitatioal field due to a uniform circular ring at a point on its axis

4.1 Gravitational potential due to a point mass
4.2 Gravitational potential due to a uniform solid sphere
4.3 Gravitational potential due to a uniform thin spherical shell
4.4 Gravitational potential due to a uniform ring at a point on its centre
4.5 Relation between gravitational field and gravitational potential

5.1 Gravitational potential energy for a system of particles
5.2 Gravitational potential energy of a body on earth's surface

6.1 Orbital speed of satellite
6.2 Time period of a satellite
6.3 Angular momentum of a satellite
6.4 Energy of a satellite
6.5 Types of satellite
6.6 Binding energy
6.7 Escape velocity
6.8 Weightlessness

7.1 Kepler's first law
7.2 Kepler's second law
7.3 Kepler's third law

The relation between gravitational field and gravitational potential is given by,

$$\overrightarrow E = - {\text{gradient }}V$$ $$\overrightarrow E = - \left[ {\frac{{\delta V}}{{\delta x}}\widehat i + \frac{{\delta V}}{{\delta y}}\widehat j + \frac{{\delta V}}{{\delta z}}\widehat k} \right]$$

where, ${\frac{{\delta V}}{{\delta x}}}$, ${\frac{{\delta V}}{{\delta y}}}$ and ${\frac{{\delta V}}{{\delta z}}}$ are the partial derivatives of $V$ wrt $x$, $y$ and $z$ respectively.

Also, we can write,

$$dV = - \overrightarrow E .\,d\overrightarrow r $$

where, $$\,d\overrightarrow r = dx\widehat i + dy\widehat j + dz\widehat k$$$$\,\overrightarrow E = {E_x}\widehat i + {E_y}\widehat j + {E_z}\widehat k$$

Note: $$dV = - \overrightarrow E .\,d\overrightarrow r $$ As, $\left( {\overrightarrow E = \frac{{\overrightarrow F }}{m}} \right)$. So, $$dV = - \frac{{\overrightarrow F .\,d\overrightarrow r }}{m}$$ or $$dV = - \frac{{dW}}{m}$$