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Illustration Electrical voltage (general definition)

Electrical voltage (general definition)
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Integration of an electric field vector \(\boldsymbol{E}\) along the line \(L\) located in an electric field:\[ U_{\text e} = \int_L \bold symbol{E} \cdot \text{d}\bold symbol{l}\where \(\text{d}\boldsymbol{l}\) is an infinitesimal line element along the considered line \(L\). The line integral of the E-field is the general definition of electric voltage \(U_{\text e}\).

The vector field \(\boldsymbol{E}\) can be divided into component \(\boldsymbol{E}_{||}\) parallel to \(\text{d}\boldsymbol{l}\) and into component \(\boldsymbol{E}_{\perp}\) orthogonal to \(\text{d}\boldsymbol{l}\). The scalar product with the orthogonal component does not contribute to the line integral: \(\boldsymbol{E}_{\perp} \cdot \text{d}\boldsymbol{l} = 0 \)