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
Answers
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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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