Question

derive an expression for gravitational potential and field due to a solid sphere at a point (i) outside the sphere (ii) on the surface of sphere (iii)inside the sphere

Answer 1

here is your solution...

hope you got it...

Answer 2

Answer:

A gravitational field can be defined as the ratio of force and mass.

Explanation:

Derivation of Gravitational potential,

dV = -∫Edr

(i) At a point outside the sphere,

dV = -∫-GM/r²dr (For limits r to ∞)

V = -GM/r

(ii) On the surface of the sphere

r = R,

V = -GM/R

(iii) inside the sphere

dV = -∫-GM/R³rdr (For the limits r to R) - GM/r

V = GM/2R³(r²) (For the limits r to R) - GM/r

V = GM/2R³(r²-R²) - GM/r

V = -GM/2R³(3R²-r²)

Similarly, Derivation of field,

E = F/m

(i) At a point outside the sphere,

E = |-GM/r²|

E = GM/r²

(ii) On the surface of the sphere

r = R,

E = GM/R²

(iii) inside the sphere

E = -GMr/R³

The opposite sign shows that the direction of force and field is towards the center of the sphere.

Hence, the gravitational potential at a point outside the sphere, on the surface of the sphere, and inside the sphere is -GM/r, -GM/R, and -GM/2R³(3R²-r²) respectively, and the gravitational field at a point outside the sphere, on the surface of the sphere, and inside the sphere are GM/r², GM/R², and -GMr/R³respectively.

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