Paper 3, Section II, A

Vector Calculus | Part IA, 2007

For a given charge distribution ρ(x,y,z)\rho(x, y, z) and divergence-free current distribution J(x,y,z)\mathbf{J}(x, y, z) (i.e. J=0)\nabla \cdot \mathbf{J}=0) in R3\mathbb{R}^{3}, the electric and magnetic fields E(x,y,z)\mathbf{E}(x, y, z) and B(x,y,z)\mathbf{B}(x, y, z) satisfy the equations

×E=0,B=0,E=ρ,×B=J\nabla \times \mathbf{E}=0, \quad \nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=\rho, \quad \nabla \times \mathbf{B}=\mathbf{J}

The radiation flux vector P\mathbf{P} is defined by P=E×B\mathbf{P}=\mathbf{E} \times \mathbf{B}. For a closed surface SS around a region VV, show using Gauss' theorem that the flux of the vector P\mathbf{P} through SS can be expressed as

SPdS=VEJdV\iint_{S} \mathbf{P} \cdot \mathbf{d} \mathbf{S}=-\iiint_{V} \mathbf{E} \cdot \mathbf{J} d V

For electric and magnetic fields given by

E(x,y,z)=(z,0,x),B(x,y,z)=(0,xy,xz)\mathbf{E}(x, y, z)=(z, 0, x), \quad \mathbf{B}(x, y, z)=(0,-x y, x z)

find the radiation flux through the quadrant of the unit spherical shell given by

x2+y2+z2=1, with 0x1,0y1,1z1x^{2}+y^{2}+z^{2}=1, \quad \text { with } \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant y \leqslant 1, \quad-1 \leqslant z \leqslant 1

[If you use (*), note that an open surface has been specified.]

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