A plane interface separates medium 1 (a dielectric of relative permittivity ) from…

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

Electricity, Magnetism and Electromagnetism, Ch Displacement current, Maxwell's equations and boundary conditions

Problem

A plane interface separates medium 1 (a dielectric of relative permittivity

\varepsilon_r = 2.0

) from medium 2 (a dielectric of relative permittivity

\varepsilon_r = 5.0

). Straddle the interface with a thin rectangular loop: two long sides of length

l = 4.0\,\mathrm{cm}

running parallel to the interface, one just inside each medium, joined by two short sides of length

h

crossing the interface. The loop is traversed in one sense and its normal

\hat{\mathbf{n}}

follows by the right-hand rule. Both fields are finite everywhere. ((a)) Write out

\oint\mathbf{E}\cdot d\mathbf{l}

round this loop in terms of the tangential field components

E_{\parallel 1}

and

E_{\parallel 2}

and

l

, explain why the contribution of the two short sides tends to zero as

h\to 0

, and use Faraday's law

\oint\mathbf{E}\cdot d\mathbf{l} = -d\Phi_B/dt

to show that

E_{\parallel 1} = E_{\parallel 2}

. ((b)) Test the limit numerically. Suppose the component of

\mathbf{B}

along

\hat{\mathbf{n}}

is changing at the very large rate

8.0\times10^{2}\,\mathrm{T\,s^{-1}}

, uniformly over the loop, and take

h = 0.10\,\mathrm{mm}

. Find the magnitude of

-d\Phi_B/dt

for this loop, and find the magnitude of the contribution

E_{\parallel}l

of one long side if the tangential field there is

300\,\mathrm{V\,m^{-1}}

. ((c)) The tangential component of

\mathbf{E}

just inside medium 1 is

300\,\mathrm{V\,m^{-1}}

. State its value just inside medium 2.

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