class27 - 1.2 Gausss law and Poissions Equation M r r2...

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1.2 Gauss’s law and Poission’s Equation Consider a point mass M , with gravitational feld at a distance r oF g = - G M r 2 ˆ r (29) We can defne a quantity oF fux through a sphere oF radius r with this point mass M in the center as ϕ =4 πr 2 g = - 4 πGM (30) we will show that the total ±ux around the surFace should be independent oF r in general. ²or an arbitrary surFace, we defne the ±ux as ϕ = ± S g · d a = ± S n · g da (31) where the integral is over the surFace S and the unit vector n n is normal to the surFace at the di³erential area da . Substituting in our expression For g and noting d a = r 2 d Ωˆ r (32) where d Ω = sin θdθdφ , and g = g ( r r (33) then ϕ = ± S gr 2 r · ˆ r ) d Ω = - GM ± S d Ω (34) where the integral over the solid angle is 4 π . Thus ϕ = ± S g · d a = - 4 πGM (35) For a point mass. We can generalize to the case where we have a discrete mass distribution ϕ = - 4 πG ² i m i (36) or a continuous distribution ϕ = - 4 πG ± V ρdv (37) 4
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class27 - 1.2 Gausss law and Poissions Equation M r r2...

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