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Illustration Electric flux and enclosed charge (1st Maxwell equation)

Electric flux and enclosed charge (1st Maxwell equation)
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The first Maxwell equation in integral form:\[ \oint_A \boldsymbol{E} ~\cdot \text{d}\boldsymbol{a} ~=~ \frac{Q}{\varepsilon_0} \]connects the charge \(Q\) trapped within the surface \(A\) with the electric flux \(\oint_A \boldsymbol{E} ~\cdot \text{d}\boldsymbol{a}\) through the surface \(A\). The scalar product \(\boldsymbol{E} ~\cdot \text{d}\boldsymbol{a}\) picks out only the portion of the electric field \(\boldsymbol{E}\) parallel to the \(\text{d}\boldsymbol{a}\) element. All other parts are eliminated in the scalar product.