50.
(a) The charge q on the capacitor as a function of time is q(t) = (
EC)(1 − e
−t/RC
), so the charging
current is i(t) = dq/dt = (
E/R)e
−t/RC
. The energy supplied by the emf is then
U =
∞
0
Ei dt =
E
2
R
∞
0
e
−t/RC
dt = C
E
2
= 2U
C
where U
C
=
1
2
C
E
2
is the energy stored in the capacitor.
(b) By directly integrating i
2
R we obtain
U
R
=
∞
0
i
2
Rdt =
E
2
R
∞
0
e
−2t/RC
dt =
1
2
C
E
2
.