Project ForgeA vector field B(x,y,z)=xî+yĵ−2zk̂ is defined over a conical region having height h=2, base radius r=3 and axis along z, with the base in the x-y plane centered at the origin. If n denotes the unit outward normal to the curved surface S of the cone, the value of the integral ∫∫_S B·n dS equals ______.
Explanation
div(B)=1+1−2=0 everywhere, so by the divergence theorem the total outward flux through the closed surface (base + curved lateral surface) is zero. On the base (z=0, outward normal −k̂), B·n=−(−2·0)=0, so the base contributes zero flux. Therefore the flux through the curved lateral surface S alone is also 0−0=0.
Source: GATE ME 2023 Official Paper (2023)
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