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Solutions to Problem Set 6
1. a) The current and charge as functions of time are
I
(
t
) =
I
0
cos
ωt
Q
(
t
) =
Q
0
sin
ωt
=
I
0
ω
sin
ωt
Electric ﬁeld points in zdirection but depends just on the radial coordinate (and time),
E
(
r,t
) =
E
(
r
)ˆ
z
sin
ωt
satisﬁes the Bessel’s diﬀerential equation
∇
2
E

1
c
2
∂
2
E
∂t
2
= 0
→
1
r
∂
∂r
r
∂E
(
r
)
∂r
+
k
2
E
(
r
) = 0
The solution is
E
(
r
) =
KJ
0
(
kr
) where
K
is some constant. We can determine it by noticing
that
Q
0
=
Z
σ dS
where
σ
=
±
0
E
(
r
)
Using also the fact that
Q
0
=
I
0
ω
we get the solution for
E
(
r
),
E
(
r
) =
I
0
2
πa±
0
c
J
0
(
kr
)
J
1
(
ka
)
In order to determine
B
(
r
) we use
∇ ×
E
=

∂
B
∂t
=
ωB
0
(
r
)ˆ
z
sin
ωt
which gives
B
0
(
r
) =
I
0
2
πac
2
±
0
J
1
(
kr
)
J
1
(
ka
)
The ﬁrst two terms in the expansions are
E
(
r
) =
I
0
πa
2
ω±
0
±
1

ω
2
r
2
4
c
2
+
ω
2
a
2
8
c
2
+
O
(
ω
4
)
²
B
(
r
) =
μ
0
I
0
r
2
πa
2
ω
±
1

ω
2
r
2
8
c
2
+
ω
2
a
2
8
c
2
+
O
(
ω
4
)
²
b) We just use the deﬁnition of
w
e
and
w
m
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