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Phys 325 Spring 2011 Lecture 9 - Physics 325 Lecture 9...

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Physics 325 Lecture 9 Gravitational Force and Potential (cont.) There are many similarities here between gravitation forces and electric forces. In fact, the only difference here is that the gravitational force is always attractive. Otherwise we just replace the masses by electric charges, and G by the appropriate electromagnetic constant and we’re done. In what follows, we carry this analogy a bit further. Given the vector g defined by Equation (8.4), we can define a gravitational flux due to a mass m through an arbitrary surface S ˆ m S S g da g nda (9.1) where ˆ n is a unit vector normal to the surface S at da . If the surface element da is a distance r away from the mass m (and outside of m ), we can write (9.1) as 2 cos m S Gm da r   where is the angle between ˆ and g n . For sim plicity let’s take the surface to be a sphere centered at m . Then 0   and the integral just gives 4 . In fact the integral will give 4 regardless of the shape of the surface since the integral is over the solid angle of the surface. Nor does it matter where inside the surface the mass m is located. The gravitational flux due to mass m is then 4 m Gm   (9.2)
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Going back to Equation (9.1) , we can use Gauss’s theorem to write ˆ m S V g nda g dV Therefore 4 4 g dV Gdm G dV     which gives 4 g G   Since g   , we find 2 4 G    (9.3) This is Poisson’s Equation.
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