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# Relation Between Gravitational Field And Potential - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• 4 Questions around this concept.

## Solve by difficulty

What is the relationship between gravitational field strength and gravitational potential?

## Concepts Covered - 1

Relation between gravitational field and potential

Gravitational field and potential are related as

$\vec{E}=-\frac{dV}{dr}$

Where E is Gravitational field

And V is Gravitational potential

And r is the position vector

And Negative sign indicates that in the direction of intensity the potential decreases.

$If \ \ \vec{r}=x \ \vec{i}+y \ \vec{j}+z \ \vec{k}$

Then $E_x=\frac{\delta V}{dx},E_y=\frac{\delta V}{dy},E_z=\frac{\delta V}{dz}$

Proof-

Let gravitational field at a point r due to a given mass distribution is E.

If a test mass m is placed inside a uniform gravitational field E.

Then force on a particle m when it is at r is  $\vec{F}=m\vec{E}$ as shown in figure

As the particle is displaced from r to r + dr the

work done by the gravitational force on it is

$dW=\vec{F}.\vec{r}=m\vec{E}.d\vec{r}$

The change in potential energy during this

displacement is

$dU=-dW=-\vec{F}.\vec{r}=-m\vec{E}.d\vec{r}$

And we know that Relation between Potential and Potential energy

As $U=mV$

So $dV=\frac{dU}{m}= -\vec{E}.d\vec{r}$

Integrating between r1, and r2

We get $V(\vec{r_2})-V(\vec{r_1})= \int_{r_1}^{r_2} -\vec{E}.d\vec{r}$

If r1=r0, is taken at the reference point, V(r0) = 0.

Then the potential V(r2=r) at any point r is

$V(\vec{r}) = \int_{r_0}^{r} -\vec{E}.d\vec{r}$

in Cartesian coordinates, we can write

$\vec{E}=E_x \ \vec{i}+E_y \ \vec{j}+E_z \ \vec{k}$

$If \ \ \vec{r}=x \ \vec{i}+y \ \vec{j}+z \ \vec{k}$

Then $d\vec{r}=dx \ \vec{i}+dy \ \vec{j}+dz \ \vec{k}$

So

$\vec{E}.d\vec{r}=-dV=E_xdx +E_ydy +E_zdz\\ dV=-E_xdx -E_ydy -E_zdz$

If y and z remain constant, dy = dz = 0

Thus $E_x=\frac{dV}{dx}$

Similarly $E_y=\frac{dV}{dy}, E_z=\frac{dV}{dz}$

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Relation between gravitational field and potential

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