ASSIGNMENT 6
Physics 218
Due Oct 22
Fall 2004
My oﬃce hours are in 114 Newman Laboratory at 1:00-3:00 PM on most Wednesdays. For
meeting at other times, please send e-mail to
dgc@lepp.cornell.edu
to request an appointment.
Appointments on Tuesdays, Thursdays, and later on Wednesdays will generally be in Wilson Lab-
oratory.
1.
A few useful electrical expressions in mks units are:
q
=
Cv
,
v
=
Ldi/dt
,
u
E
=
Cv
2
/
2,
u
M
=
Li
2
/
2 and the expressions for inductance
L
1
and capacitance
C
1
per unit length given in
Prob. 3. The units of quantities in these expressions are: [
q
] = C (Coulombs), [
v
] = V (Volts),
[
C
] = F (Farads), [
L
] = H (Henrys), [
i
] = A = C/s (Amps=Coulomb/second), and [
u
] = J/m
3
(Joules/meter
3
), and [
v/i
] = Ω (Ohms). Use these and/or other simple expressions involving
electrical quantities to show that the units of 1
/
√
L
1
C
1
and 1
√
µ
0
²
0
are m/s (velocity) and that
the units of
q
L
1
/C
1
and
q
µ
0
/²
0
are Ω (Ohms-resistance).
L
R
G
1
∆
x
C
1
∆
x
1
∆
x
1
∆
x
1
2.
In the lecture we derived di±erential equations for electromagnetic waves on a transmission
line that had resistance in the conductors as well as conductance in the insulator between the
two conductors. We used the parameters
L
1
,
C
1
,
R
1
, and
G
1
for the inductance, capacitance,
resistance, and conductance (per unit length) respectively. Then we found:
∂v
(
x, t
)
∂x
=
−
L
1
∂i
(
x, t
)
∂t
−
R
1
i
(
x, t
)
∂i
(
x, t
)
∂x
=
−
C
1
∂v
(
x, t
)
∂t
−
G
1
v
(
x, t
)
(a)
Show that these two equations imply the “telegrapher’s equation”:
∂
2
v
(
x, t
)
∂x
2
=
L
1
C
1
∂
2
v
(
x, t
)
∂t
2
+(
R
1
C
1
+
G
1
L
1
)
∂v
(
x, t
)
∂t
+
R
1
G
1
v
(
x, t
)
(b)
What wave equation does
i
(
x, t
) satisfy?
(c)
Assume solutions of the form
v
(
x, t
)=
V
0
e
jωt
+
γx
and
i
(
x, t
)=
I
0
e
jωt
+
γx
. Using the original
equations or the equations in Parts (a) and (b), ²nd a (complex) expression for
γ
2
in terms
of the given parameters and
ω
. (Remember
j
=
√
−
1 in the electrical engineering convention