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Gauss's Law - for both Electricity and Gravity?



This past week in physics, we learned about Gauss's Law for electricity. It states that the electric flux, or the amount of electric field penetrating a surface, is proportional to the charge enclosed within the surface. Interestingly, Gauss's Law does not only apply to electricity: it also applies to gravity. According to Wikipedia, gravitational flux is a surface integral of the gravitational field over a closed surface. This is analogous to electric flux, equivalent to the surface integral of the electric field over a closed surface. Gauss's Law for gravity is mathematically represented by this equation:  \oiint{\displaystyle \scriptstyle \partial V}\scriptstyle \partial V {\displaystyle \mathbf {g} \cdot d\mathbf {A} =-4\pi GM}\mathbf {g} \cdot d\mathbf {A} =-4\pi GM , where \oiint represents a surface integral over a closed surface. Gauss's Law for electric fields states that:

\Phi_E = \frac{Q}{\varepsilon_0} = \oiint{\displaystyle \scriptstyle _{S}}{\displaystyle \scriptstyle _{S}} {\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {A} }\mathbf{E} \cdot \mathrm{d}\mathbf{A} . Electric flux can also be represented by 4 pi k Q. Since G is the gravitational constant analogous to k for electricity, and since M is analogous to charge, it makes sense that total gravitational flux is equivalent to -4 pi GM. Gravitational flux is negative because gravitation fields always attract, where electric flux can be positive or negative depending on the enclosed charge.

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Looks like this needs a little bit more work on the formatting, but it is interesting that these two are very similar to each other.

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