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

.