Slate is a sedimentary rock that can be split horizontally to produce smooth flat surfaces that is often used as a roofing material.
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Yes, the micas, which are aluminum silicates, separate into very thin, smooth layers.
If the field lines were not perpendicular to the surface, then they could be decomposed into components perpendicular and parallel to the surface. But if there is an E-field along the surface, the surface is no longer an equipotential.
Along the height it is hb^3/48 and along base it is bh^3/36
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