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My question is this: Are all solutions to Poisson's equation for a certain charge density such that the negative gradient of them will give me the electric field?

In deriving the equation, we say that every potential (function whose negative gradient is E due to charge density) satisfies Poisson's equation (using one of Maxwell's equations), but does every solution to Poisson's equation correspond to a function whose negative gradient is the electric field due to that charge density?

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If we have a function $V$ that solves Poisson's equation, then it must be true that $$\nabla^2V=\nabla\cdot(\nabla V)=-\frac{\rho}{\epsilon_0}$$

By definition of potential, we have $\nabla V=-\mathbf E$, so then $$\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0}$$

Therefore, the answer to

does every solution to Poisson's equation correspond to a function whose negative gradient is the electric field due to that charge density?

is yes. Just take the gradient of the solution $V$. By definition, that is the field caused by the charge distribution. The result is called Gauss's law.

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