How can you prove that dS in φ =E.dS is dS Cos θ if the area is tilted at an angle ? I need the mathematical Steps.
Electric Flux is defined as the total no of field lines passing normal to the surface. So while calculating, we need to consider the area of the surface normal to the field lines only. That is why we take the dot product of E and ds, where ds is the area vector (not just the area: Area vector is a vector whose magnitude is equal to the area and dirtected normal to the surface.)
Then by definition of dot product,
dφ =E.dS = EdS cos θ
which gives the component of E and the component of area vector in the direction of E (When area vector is in the direction of E, the actual component of area is perpendicular to E)
Hope the matter is clear.
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included quadratic curvature terms but no cosmological constant. Numerical solutions were found in 4 + 1 dimensions in the presence of a cosmological constant and a scalar field, along with the quadratic curvature terms. In this latter scenario, metrics exist that depend periodically on the extra spatial coordinate so that choosing the size of the extra dimension to be the period produces a compact extra dimension without any fine-tunings or singularities. The parameters in the action ¯x the size of the extra dimension uniquely.
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