In a region of a laboratory the electric field is measured to be E = (2axy + bz

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Gradient, divergence and curl

Problem

In a region of a laboratory the electric field is measured to be E=(2axy+bz, ax2, bx),a=3.0 V m−3,b=5.0 V m−2,\mathbf E = (2axy + bz,\ ax^2,\ bx),\qquad a = 3.0\ \mathrm{V\,m^{-3}},\quad b = 5.0\ \mathrm{V\,m^{-2}}, with xx, yy, zz in metres. ((a)) Show that ∇×E=0\nabla\times\mathbf E = \mathbf 0. ((b)) Construct the potential φ\varphi with E=−∇φ\mathbf E = -\nabla\varphi and φ=0\varphi = 0 at the origin, and find φ\varphi at P=(1.0, 2.0, 1.0) mP = (1.0,\ 2.0,\ 1.0)\ \mathrm m. ((c)) A second field is G=c (y, 2x, 0)\mathbf G = c\,(y,\ 2x,\ 0) with c=3.0 V m−2c = 3.0\ \mathrm{V\,m^{-2}}. Find ∇×G\nabla\times\mathbf G, and show where the construction of (b) breaks down for it.

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