Problem 4 : Two concentric spherical shells of mass “
m
1
m
1
” and “
m
2
m
2
” have radii “
r
1
r
1
” and “
r
2
r
2
” respectively, where
r
2
r
2
>
r
1
r
1
. Find gravitational intensity at a point, which is at a distance “r” from the common center for following situations, when it lies (i) inside smaller shell (ii) in between two shells and (iii) outside outer shell.

Solution : Three points “A”, “B” and “C” corresponding to three given situations in the question are shown in the figure :

The point inside smaller shell is also inside outer shell. The gravitational field inside a shell is zero. Hence, net gravitational field at a position inside the smaller shell is zero,

E
1
=
0
E
1
=
0

The gravitational field strength due to outer shell (
E
o
E
o
) at a point inside is zero. On the other hand, gravitational field strength due to inner shell (
E
i
E
i
) at a point outside is :

⇒
E
i
=
G
M
r
2
⇒
E
i
=
G
M
r
2

Hence, net gravitational field at position in between two shells is :

E
2
=
E
i
+
E
o
=
G
m
1
r
2
E
2
=
E
i
+
E
o
=
G
m
1
r
2

A point outside outer shell is also outside inner shell. Hence, net field strength at a position outside outer shell is :

E
3
=
E
i
+
E
o
=
G
m
1
r
2
+
G
m
2
r
2
E
3
=
E
i
+
E
o
=
G
m
1
r
2
+
G
m
2
r
2

⇒
E
3
=
G
m
1
+
m
2
r
2
⇒
E
3
=
G
m
1
+
m
2
r
2