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Show that the gravitational field inside the cavity is uniform(same magnitude and direction) and find its value

Show that the gravitational field inside the cavity is uniform(same magnitude and direction) and find its value

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Grade:11

1 Answers

Arun
25750 Points
6 years ago

The gravitational field produced by a sphere of uniform density rho at a radius r inside the sphere is radially downward and has magnitude F = Kr where K = G(4/3)pi rho = (GM/R^3). This follows from the fact that the field is the same as if all the mass from 0 to r (e.g. rho * (4/3) pi r^3) were concentrated at the center. Of course the mass above r doesn't contribute. This can also be written as a vector equation F = -Kr where r is the radius vector from the center to the point. 

Now consider any point P within the cavity. Let r be the vector displacement from the center of the big sphere to P. Let S be the vector displacement from the center of the big sphere to the center of the cavity (|S| = R/2). 

The field at P is the same as the field produced by a solid sphere of radius R minus the field produced by the smaller (removed) sphere of radius R/2. The displacements of P from the centers of the two spheres are respectively: r and r-S, so the total field is: 

F = -Kr-(-K(r-S)) = -KS 

In other words the gravitational field everywhere in the cavity is parallel to the axis connecting the center of the cavity to the center of the big sphere, and has the same magnitude (KR/2) as the surface gravity of a solid sphere of radius R/2. Another way of putting it is that the gravity is the same as it would be at the center of the cavity if there were no cavity.

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