$$ CALC A capacitor with capacitance

C

is connected in series to a resistor of resistance

R

and a battery with emf

$.$ The circuit is completed at time

t

$=$

$0$

$.($a$)$ In terms of

$,$

R

$,$ and

C

$,$ how much energy is stored in the capacitor when it is fully charged? $($b$)$ The power output of the battery is

P

$=$

i

$,$ with

i

given by Eq$.(26\mathrm{.}13).$ The electrical energy supplied in an infinitesimal time interval

d

t

is

P

d

t

$.$ Integrate from

t

$=$

$0$

to

t

$$

to find the total energy supplied by the battery. $($c$)$ The rate of consumption of electrical energy in the resistor is

P

R

$=$

i

$2$

R

$.$ In an infinitesimal time interval

d

t

$,$ the amount of electrical energy consumed by the resistor is

P

R

d

t

$.$ Integrate from

t

$=$

$0$

to

t

$$

to find the total energy consumed by the resistor. $($d$)$ What fraction of the total energy supplied by the battery is stored in the capacitor? What fraction is consumed in the resistor?