See the picture,

                     Consider the Grey sphere to be the big sphere and the yellow sphere to be smaller one with negative density( 1. We are applying superposition principle . 2. So if u superimpose the 2 u shud get the sphere on the LHS and hence it must have neg.density)

 

 

Now within the first sphere i have made dotted sphere of radius l from the center, from which we can calculate the gravitational field using the formula for a distance l.

 

Now do that for other sphere.(but at any other pt).

 

Now vector addn , u 'll get a constant value. A variant of this in electrostatics was asked in 2007 paper btw.

 

I 'll put the ans tomm. ,need to go to sleep. (sorry , but u can get it from this pt)

It's simple

 

See  E = GM/ r^3  r

            = 4/3 pi  r .......................... ( 1 )

( M is the mass enclosed )

      fist suppose that the cavity was not there . And we are interested in finding E at the point p from the centre of the cavity . The point that we are considering is at r from the centre of the big sphere .

 

So we can write

                               r = p + l ........................ ( 2 ) 

 now we can write that

 

          E ( due to solid sphere ) = E ( due to sphere with its cavity ) + E ( due to solid cavity )

 

so,  E ( due to sphere with its cavity )

 

 =            E ( due to solid sphere ) - E ( due to solid cavity ) .............. ( 3 )

 

Now using ( 1 ) & ( 3 )

 

    we have reqd E = 4/3 pi  ( r - p )

                             = 4/3 pi  l         ( from ( 2 ) )