Gauss's Law vs Newton's Law

Question

This is thought experiment. I couldn't get a good answer because I keep getting negative mass.

Gauss's Law say that eletric field is proportional to charge, how much charged is enclosed. Newton's gravitational pretty much says the same thing. More mass, strong gravitational strength.

So to put it mathematically

$$\vec{E} \propto \sum Q_{en}$$

But charge can be negative, a negative sum of charge means the electric field is going inwards.

$$\vec{g} \propto \sum M_{en}$$

But mass can only be positive, but g is then propotional to the mass enclosed, which means it will go radially outwards. That doesn't make sense because we know gravity is radially inward for a spherical shape (like earth)

Answer 1

In the equations as you've written them, the constant of proportionality is an outward-pointing vector for the electric field and an inward-pointing vector for the gravitational field. Or in other words, if you take the radial component only: it's a positive constant for the electric force and a negative constant for the gravitational force.

The details: Gauss's law for electrostatics actually says

$$\iint\vec{E}\cdot\mathrm{d}\vec{A} = \frac{Q_\text{enc}}{\epsilon_0}$$

and for Newtonian gravity, you can write

$$\iint\vec{g}\cdot\mathrm{d}\vec{A} = -4\pi G M_\text{enc}$$

For a spherically symmetric surface and mass/charge distribution, letting $\hat{n}$ represent the outward-pointing normal vector at each point on the surface, these simplify to

$$\vec{E} = \frac{Q_\text{enc}}{4\pi\epsilon_0 r^2}\hat{n}$$

and

$$\vec{g} = -\frac{GM_\text{enc}}{r^2}\hat{n}$$

Note that the constant of proportionality in the first case is $\hat{n}/4\pi\epsilon_0 r^2$, which points outward, and in the second case is $-G\hat{n}/r^2$, which points inwards.