By Edwin Bidwell Wilson
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Extra resources for A Statistical Discussion of Sets of Precise Astronomical Measurements IV
The product of jzext and the cross-section area of the cylinder is ﬁnite. It is called the line current density and has the only component Iz , independent of the cross-section shape. The ﬁelds are inﬁnite near the line current. They are found from the Maxwell equations in which jz is proportional to δ(r ), so that the integral of jz over any part of the surface z = const containing the point r = 0, is ﬁnite and equal to Izext . The ﬁelds depend neither on z nor on the azimuthal coordinate ϕ, and have only the components H ϕ and Ez .
The amplitudes of these waves are found from the demand that the sum of their ﬁelds equals the given ﬁeld. 19), both these amplitudes are equal to 1/2. The ﬁelds of the eigenwaves at z > 0 are obtained from the ﬁelds at z = 0 by multiplying by exp(−ihrg z) and exp(−ihlf z), respectively, so that the resulting ﬁeld has the components 1 [exp(−ihpg z) + exp(−ihlf z)], 2 i Ey (z) = [exp(−ihpg z) − exp(−ihlf z)]. 2 √ Since hrg + hlf = 2k εµ, hrg − hlf = −2kκ , we have Ex ( z ) = √ Ex (z) = exp(−ik εµz) cos(κ z), √ Ey (z) = exp(−ik εµz) sin(κ z).
82) are necessary and sufﬁcient for the line on which some components of the ﬁelds become inﬁnite, not to be a linear source, but an edge. Below we show that if these conditions hold, then the energy ﬂux out of the line is zero. 82) the equality to zero of the energy ﬂux is only necessary (but not sufﬁcient) for the line to be an edge. In certain cases it is valid for a linear source, too. 82) violates. 82) holds, we ﬁnd the ﬁeld structure near the edge. 4). Align the z-axis of the cylindrical coordinate system along the wedge edge.