Question

In a certain region of space gravitational field is given by I = - (k//r). Taking the reference point to be at r = r_(0), with gravitational potential V = V_(0), find the gravitational potential at distance r.

Answer 1

Given:

In a certain region of space the Gravitational Field Intensity is given by I = - (k/r). The reference point is considered to be at r = r_(0) with Gravitational Potential V = V_(0).

To find:

Gravitational Potential at a distance r

Calculation:

In an Gravitational Field , the basic relationship between Gravitational Field Intensity (vector) and Gravitational Potential (scalar) can be said as :

$\therefore \: \: dV = I \times dr$

Integrating on both sides :

$= > \displaystyle \int \:dV = \int I \times dr$

Putting the limits:

$= > \displaystyle \int_{V_{0}}^{V} \:dV = \int_{r_{0}}^{r} I \times dr$

$= > \displaystyle \int_{V_{0}}^{V} \:dV = \int_{r_{0}}^{r} \bigg( - \dfrac{k}{r} \bigg ) \times dr$

$= > \displaystyle \int_{V_{0}}^{V} \:dV = - k\int_{r_{0}}^{r} \dfrac{dr}{r}$

$= > V - V_{0} = - k \bigg \{ ln(r) - ln(r_{0}) \bigg \}$

$= > V - V_{0} = - k \bigg \{ ln \bigg( \dfrac{r}{r_{0}} \bigg) \bigg \}$

$= > V = V_{0} - k \bigg \{ ln \bigg( \dfrac{r}{r_{0}} \bigg) \bigg \}$

So final answer is :

Potential at distance r from reference is :

$\boxed{ \red{ \bold{ V = V_{0} - k \bigg \{ ln \bigg( \dfrac{r}{r_{0}} \bigg) \bigg \}}}}$