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Gravitational Potential

At a point in a gravitational field, potential V is defined as negative of work done per unit mass in shifting a test mass from some reference point (usually at infinity) to the given point i.e.,
46350.png46344.png[as F/m = I]
 
∴ 46331.png
 
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 indicates that the direction of intensity is in the direction where the potential decreases.
 
∴ Gravitational potential produced by a point mass 46325.png

Gravitational Potential for Different Bodies

Potential due to uniform ring
 
At a point on its axis
At the center
46313.png
46307.png
 
46319.png
Fig. 9

Potential due to spherical shell
 
Outside the surface
r > R
On the surface
r = R
Inside the surface
r < R
46287.png
46281.png
46275.png
 
46294.png
Fig. 10
 

​Potential due to uniform solid sphere
 

Outside the surface

r > R

On the surface

r = R

Inside the surface

r < R

46259.png

 

46247.png

46241.png

 

At the center (r = 0)

46235.png

 
46253.png
Fig. 11




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