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Illustration Electric flux - scalar product of surface orthogonal vector and E-field

Electric flux - Scalar product of surface vector and E-field
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For the First Maxwell equation you can use any closed volume, like in this case a cube volume. The electric flux \( \Phi \) through the infinitesimal surface element of the cube is defined as the scalar product between the electric field vector \(\boldsymbol{E}\) at the considered surface point and the surface orthogonal vector \(\text{d}\boldsymbol{a}\). \(\text{d}\boldsymbol{a}\) is orthogonal to the considered surface element and its magnitude represents the area of the surface element:\[ \text{d}\Phi = \boldsymbol{E} \cdot \text{d}\boldsymbol{a} \]

The electric field vector \(\boldsymbol{E}\) can be divided into a component \(\boldsymbol{E}_{\perp} \) orthogonal to \(\text{d}\boldsymbol{a}\) and into a component \(\boldsymbol{E}_{||} \) parallel to \(\text{d}\boldsymbol{a}\). The scalar product with the orthogonal component is always zero.