A hemispherical bowl of radius has its flat rim lying in the plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Line, surface and volume integrals

Problem

A hemispherical bowl of radius R=0.4 mR = 0.4\ \mathrm{m} has its flat rim lying in the xyxy plane, centred on the origin, and its dome rising above the plane (z>0z > 0). Someone looking straight down at the bowl from a point high on the +z+z axis sees the rim traversed anticlockwise. ((a)) Using the right-hand rule for an open surface and its boundary, state the direction of dS\mathrm{d}\mathbf{S} (in terms of the local radial unit vector r^\hat{\mathbf{r}} of spherical polars) implied by this rim direction, and write down dS\mathrm{d}\mathbf{S} in spherical polar coordinates on the dome. ((b)) A uniform field E=300 V m−1 z^\mathbf{E} = 300\ \mathrm{V\,m^{-1}}\,\hat{\mathbf{z}} fills the region. Find the flux of E\mathbf{E} through the dome, using the orientation found in part (a).

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