An infinite flat sheet lies in the plane

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Maxwell's equations and conservation laws

The Boundary-Value Problem, Ch The vector potential and the magnetostatic boundary-value problem

Problem

An infinite flat sheet lies in the

z=0

plane. It carries a uniform free surface charge density

\sigma_f=3.0\times10^{-8}\,\mathrm{C\,m^{-2}}

and, independently, a uniform surface current density

\mathbf{K}=6.0\,\mathrm{A\,m^{-1}}\,\hat{\mathbf{y}}

. Both half-spaces (

z>0

and

z<0

) are vacuum, with

\hat{\mathbf{n}}_{12}=\hat{\mathbf{z}}

pointing from the

z<0

side (medium 1) into the

z>0

side (medium 2). ((a)) Recall the interface condition on

V

across a charged sheet (met earlier): state the jump in

E_z=-\partial V/\partial z

across the sheet, and its numerical value. ((b)) State the analogous interface condition on

\mathbf{A}

that carries the same information for a surface current, and compute the jump in

\partial A_y/\partial z

across the sheet due to

\mathbf{K}

. ((c)) Say, in one sentence for each, what stays continuous and what jumps for

V

and for

\mathbf{A}

at this sheet, and why the pattern is the same in both cases.

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