648
14 Electromagnetism II
reflected and transmitted waves are denoted by E I , E R , E T , k I , k R and k T
respectively. The wave is incident onto the interface at the origin and makes
an angle θ to the normal. Both incident and reflected waves propagate in the
medium with refractive index n 1 . The transmitted wave propagates in the
medium with refractive index n 2 at refraction angle φ.
Use the boundary conditions satisfied by the electric field at the interface to
show that the reflectance, R, is given by
R =
n 2 cos θ − n 1 cos φ
n 2 cos θ + n 1 cos φ
2
14.81 (a) Using the expressions for R in prob. (14.80) and Snell’s law, show that
the reflectance is zero when tan θ =
n 2
n 1
(b) Calculate this angle for electromagnetic radiation in air incident onto
glass, which has a refractive index n = 1.5.
14.82 (a) For normal incidence the reflection coefficient, R, at the planar surface
between two dielectric media is given by
R =
(n 1 − n 2 ) 2
(n 1 + n 2 ) 2
where n 1 and n 2 are the refractive indices of the media. Sketch the form
of R against n 1 /n 2 . On the same figure sketch the form of T against
n 1 /n 2 where T is the transmission coefficient. Indicate numerical values
of T and R where appropriate.
(b) At what value of n 1 /n 2 does R = T ?
[University of Durham 2000]
14.83 Consider the solutions of linearly polarized harmonic plane waves
E = E 0 e
i(ωt−k.r +ϕ)
, B = B 0 e
i(ωt−k.r +ϕ)
where E 0 and B 0 are constant vectors associated with maximum amplitude
of oscillations. Show that (a) B is perpendicular to E; (b) B is in phase with
E and (c) the magnitudes of B and E are related by B = E/c for free space
in the SI system.
14.84 An uncharged dielectric cube of material of relative permittivity 6 contains
a uniform electric field E of 2 kV/m, which is perpendicular to one of the
faces. What is the surface charge density induced on this face?
[University of Manchester 2006]
14 Electromagnetism II
reflected and transmitted waves are denoted by E I , E R , E T , k I , k R and k T
respectively. The wave is incident onto the interface at the origin and makes
an angle θ to the normal. Both incident and reflected waves propagate in the
medium with refractive index n 1 . The transmitted wave propagates in the
medium with refractive index n 2 at refraction angle φ.
Use the boundary conditions satisfied by the electric field at the interface to
show that the reflectance, R, is given by
R =
n 2 cos θ − n 1 cos φ
n 2 cos θ + n 1 cos φ
2
14.81 (a) Using the expressions for R in prob. (14.80) and Snell’s law, show that
the reflectance is zero when tan θ =
n 2
n 1
(b) Calculate this angle for electromagnetic radiation in air incident onto
glass, which has a refractive index n = 1.5.
14.82 (a) For normal incidence the reflection coefficient, R, at the planar surface
between two dielectric media is given by
R =
(n 1 − n 2 ) 2
(n 1 + n 2 ) 2
where n 1 and n 2 are the refractive indices of the media. Sketch the form
of R against n 1 /n 2 . On the same figure sketch the form of T against
n 1 /n 2 where T is the transmission coefficient. Indicate numerical values
of T and R where appropriate.
(b) At what value of n 1 /n 2 does R = T ?
[University of Durham 2000]
14.83 Consider the solutions of linearly polarized harmonic plane waves
E = E 0 e
i(ωt−k.r +ϕ)
, B = B 0 e
i(ωt−k.r +ϕ)
where E 0 and B 0 are constant vectors associated with maximum amplitude
of oscillations. Show that (a) B is perpendicular to E; (b) B is in phase with
E and (c) the magnitudes of B and E are related by B = E/c for free space
in the SI system.
14.84 An uncharged dielectric cube of material of relative permittivity 6 contains
a uniform electric field E of 2 kV/m, which is perpendicular to one of the
faces. What is the surface charge density induced on this face?
[University of Manchester 2006]