Phys3320(HW8,(due(Thurs(night,(Mar(20(2025.(((Optional(rework(due(by(Sat(noon)(
1.
A) Starting with Maxwell’s equation in matter (in terms of D and H fields) show that, for a
linear homogeneous dielectric (D= eE, B= µH) with no free charges or currents (rfree=0, Jfree =0),
the E and B fields each obey a wave equation, with a common wave speed given by
𝑣 = 1/
√
𝜀𝜇
.
B) Starting from the same Maxwell equations as in part A, rewrite them in integral form, and
then briefly sketch out for us the reasoning which leads to all the boundary conditions on E and
B at a planar interface between two different linear materials (labeled 1 and 2), with
permittivities and permeabilities e1, µ1 and e2, µ2, respectively. (Again, assume no free charge or
current densities)
C) Suppose I tell you
𝐴𝑒!"# + 𝐵𝑒!$# = 𝐶𝑒!%# ,.
(with nonzero constants
𝐴, 𝐵, 𝐶, 𝑎 , 𝑏, 𝑐
) is true for
all times t. Prove that
𝑎 = 𝑏 = 𝑐
, and also that
𝐴 + 𝐵 = 𝐶
2.
In Griffiths’ section 9.3.2, (Reflection and transmission at normal incidence) he finds
reflection and transmission coefficients (R and T). But he made the assumption that µ1 = µ2 = µ0.
Drop that assumption, i.e. keep µ1 and µ2 general, and find the general formulas for R and T.
To check, explicitly confirm that R+T=1, still (as it must be)
Hint: Don’t redo work Griffiths has done for you. Use whatever you need from section 9.3.2, just
be careful not to use results where he has assumed µ1 = µ2 = µ0. I claim you can express your
final results for R and T purely as very simple functions of
b
only!
3.
In my lecture notes (and/or Griffiths 9.3.3) we worked out the case of reflection and
transmission at any angle. But we considered the case where the incident E-field is polarized in
the plane of incidence. Go through that section again, but work out the different case where the
E-field is polarized perpendicular to the plane of incidence. (You should once again assume
µ1=µ2=µ0.) Thus, start with
𝐸
3
4
5
!&
(
𝑟
4
, 𝑡
)
=. 𝐸
5
',!&𝑒! (*
+
,
∙.
,
/0#)𝑦
; and
𝑘
3
4
!& = 𝑘!& sin 𝜃!& 𝑥
;
+.𝑘!& cos 𝜃!& 𝑧
E
Specifically, what I mean by “work out” is:
A) Make a clear sketch (modeled on Griffiths fig 9.15) of the geometry and angles for this case.
Then, write out what the four boundary conditions become in this case (i.e. modify Griffiths Eq
9.102 through 9.105 appropriately for this new situation).
Finally, find the new “Fresnel Equations”, i.e. a version of Eq 9.109, but for this polarization
case. Explicitly check that your Fresnel equations reduce to proper results at normal incidence.
(8 pts for this part)
B) Replicate Griffiths Figure 9.16, (but of course for this perpendicular polarization case.)
Use Mathematica (or some program) please, don’t just “sketch it”. Assume n2/n1=2.0
Briefly, discuss what is similar, and what is different, about this case from what Griffiths (and I)
solved. Is there a “Brewster’s angle” for your situation, i.e. a non-trivial angle where reflection
becomes zero?
C) Replicate Griffiths Figure 9.17 (the one at the end of 9.3) but again, for this perpendicular
polarization case, and again assuming n2/n1=2.0 and again using a computer to plot. Show that
R+T=1 for this situation, no matter what the angle. Briefly, comment on the physics!
(HW 8 continued on back )
