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Physics 2214 Problem Set #8
(Due 11:15 am, Thursday 10/30/07)
1.
YF derives the (1D) wave equation (on pp. 10991101) by assuming a plane wave
propagating in the
x
direction that is polarized in the
y
direction.
The second part of the
derivation is based on applying Ampere’s law counterclockwise around rectangular loop
efghe
in the
xz
plane (Fig. 32.11, Eq. 32.13). We’re going to generalize this approach to do a
major step in the derivation of the differential form of Ampere’s law from the integral form.
(a)
Generalize Eq. 32.13 by calculating
d
⋅
∫
Bl
G
G
v
around the same loop but
without
any
assumptions about the direction of
B
G
or the direction of propagation.
(In other words, let
B
G
have
x
,
y
, and
z
components and let it depend on
x
,
y
,
z
, and
t
.) Substitute
Δ
z
for their
a
.
(b) Divide
d
⋅
∫
GG
v
by the area
Δ
x
Δ
z
and take the limits
Δ
x
→
0 and
Δ
z
→
0 to show that the
circulation per unit area is the
y
component of the curl of
B
G
;
i.e.
show that
()
,0
lim
y
xz
d
ΔΔ→
⋅
=∇×
ΔΔ
∫
B
G
G
G
v
[We won’t do the rest of the derivation, but it goes like this:
Applying Ampere’s law to this
loop yields
() ()
00
/
y
y
Et
εμ
∇×
=
∂
∂
B
G
. We can then take loops in the
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 Fall '07
 GIAMBATTISTA,A
 Physics, Ampere, 3D wave equations

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