Using Newtonian field equation to find the gravity inside a sphere

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Homework Statement



work out g field at a distance r=R/2 from the centre of a spherically symmetric body of radius R.

Homework Equations



[itex]\nabla^{2}\psi=4\pi G \rho[/itex]

[itex]\psi=G\int_{V'}\rho\frac{1}{|\vec{r}-\vec{r^{'}}|}d^{3}\vec{r^{'}}[/itex]

[itex]-\int_{V}\nabla\cdot g dV = \int_{V}\rho dV[/itex]

V=4/3∏r3

The Attempt at a Solution



V=4/3∏r3

V1=4/3∏(R/2)3 = 1/6∏R3
V2=4/3∏R3
[itex]\int_{1/6\pi R^{3}}^{4/3\pi R^{3}}\rho dV[/itex]

[itex]=\rho(4/3\pi R^{3}-1/6\pi R^{3})[/itex]

[itex]=\frac{7}{6}\rho \pi r^{3}[/itex]
 

Answers and Replies

  • #2
82
0

Homework Statement



work out g field at a distance r=R/2 from the centre of a spherically symmetric body of radius R.

Homework Equations



[itex]\nabla^{2}\psi=4\pi G \rho[/itex]

[itex]\psi=G\int_{V'}\rho\frac{1}{|\vec{r}-\vec{r^{'}}|}d^{3}\vec{r^{'}}[/itex]

[itex]-\int_{V}\nabla\cdot g dV = \int_{V}\rho dV[/itex]

V=4/3∏r3

The Attempt at a Solution



V=4/3∏r3

V1=4/3∏(R/2)3 = 1/6∏R3
V2=4/3∏R3
[itex]\int_{1/6\pi R^{3}}^{4/3\pi R^{3}}\rho dV[/itex]

[itex]=\rho(4/3\pi R^{3}-1/6\pi R^{3})[/itex]

[itex]=\frac{7}{6}\rho \pi r^{3}[/itex]

should be a G*2*pi in the front of one of those integrals
 

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