Ok right here we go.

There is a theorem in mathematics called stokes theorem which like all the best maths is bloody obvious when you think about it but has profound consequences.

This says that for a substance that is conserved

Inside a closed surface - one that joins up like a sphere

**The sum of the amount produced - the sum of the amount destroyed = the amount going through the surface.**If you think of the stuff as water this is saying, it has to go somewhere.

Gravitational field is a conservative field - it is conserved

and it is created by mass so we can rewrite the above as:

**Sum of gravitational flux over a surface = Mass within this surface.**gravitational flux is just the field times the area you are looking at.

or in the terms of the above diagram - where

*g* = the gravitational field at a point

*ds* = the bit of surface you ar looking at

sum the flux over the surface = sum of

*g* x dS over the surface

M

there is a constant in there we will call

*G* the gravitational constant

s

Now if we consider a sphere everything is beautifully symmetric

so the field must be symmetric so g must be constant.

This is great because it means all we need to know to work out the gravitational field produced by a uniformly distributed mass in a sphere is it's Mass and the surface area

* g Total mass/ Surface area*the area of a sphere is

*4πr*^{2} so:

* g M / 4πr*^{2}Oh look if M is constant eg if we are outside the body we appear to have derived the inverse square law, which is reassuring.

**This means that outside of a spherical body we can consider the field produced by it is identical to a point mass** - which makes calculations far far easier.

Now if we consider a hollow sphere:

the field produced by the shell inside it inside must again be uniform:

*Now the flux leaving outwards + the flux leaving inwards = Mass of the shell*We can work out the flux leaving outwards:

We worked out earler that a spherically symmetric body's field is the sum of it's symmetrically symmetric parts so:

*flux leaving a solid sphere = flux leaving a sphere the size of the hole + affect of the shell*as the flux leaving the shell inwards will make no difference to the big shpere

*flux*_{solid} = flux_{hole} + flux leaving shell outwardsthis can be rearranged to:

*flux leaving the shell outwards = flux*_{solid} - flux_{hole}*flux leaving the shell outwards M*_{solid} - M_{hole} We know that the

*mass of the shell = mass of a solid sphere - mass of a sphere the size of the hole*So

*Total flux leaving the shell mass of a solid sphere - mass of a sphere the size of the hole**Total flux leaving the shell = flux leaving the shell outwards***so there is none left to leave inwards so the field inside the hollow spherical shell must be zero!**I hope that makes some sense I apologise for using = &

interchageably, if you did it all properly it would have the same result