# Gravitational Potential

## Gravitational Potential

At a point on a gravitational field potential V is defined as negative of work done per unit mass in shifting a test mass from some reference point to the given point i.e.,

$$V=-\frac{W}{m}=-\int{\frac{\overrightarrow{F}.\overrightarrow{dr}}{m}}=-\int{I.dr}$$           $$\left( As\,\,\frac{F}{m}=I \right)$$     i.e,  $$\therefore \,\,I=-\frac{dV}{dr}$$

i.e. negative gradient of potential gives intensity of field or potential is a scalar function of position whose space derivative gives intensity. Negative sign indicated that the direction of intensity is in the direction where the potential decreases.

i) It is a scalar quantity because it is defined as work done per unit mass.

ii) Unit: $$Joule/kg$$ (or) $${{m}^{2}}/{{\sec }^{2}}$$ and Dimension: $$\left[ {{M}^{0}}{{L}^{2}}{{T}^{-2}} \right]$$

iv) If the field is produced by a point mass then, $$V=-\int{I.dr}=-\int{\left( -\frac{GM}{{{r}^{2}}} \right)}dr$$   $$\left[ As\,\,I=-\frac{GM}{{{r}^{2}}} \right]$$

$$\therefore \,\,V=-\frac{GM}{r}+c$$     [Here, c = Constant of integration]

Assuming reference point at $$\infty$$ and potential to be zero there we get: $$0=-\frac{GM}{\infty }+c$$$$\Rightarrow c=0$$

$$\therefore \,$$ Gravitational potential $$\left( \,V \right)=-\frac{GM}{r}$$

v) Gravitational potential difference: It is defined as the work done to move a unit mass from one point to the other in the gravitational filed. The gravitational potential difference in bringing unit test mass m from point A to point B under the gravitational influence of source mass M is,

$$\Delta V={{V}_{A}}-{{V}_{B}}=\frac{{{W}_{A\to B}}}{m}=-GM\left( \frac{1}{{{r}_{A}}}-\frac{1}{{{r}_{B}}} \right)$$.