Question

A rechargeable battery has emf ε_battery​=14.0 V, internal resistance r=4.00 Ω and battery capacity (total charge) 2.00 Ah. 

When the battery is fully charged, a light bulb is connected to it for 8 hours (no other elements are in the circuit). The light bulb has specs 14.0 V and 1.00 W.

We assume the light bulb has constant resistance (independent of the current) and the emf of the battery remains constant.

After these 8 hours the battery is disconnected from the light bulb and connected in this circuit for charging:

Picture: +ε to +battery to -battery to resistor R₁ to -ε

The charger consists of a ideal voltage source with ε=20.0 V (with no internal resistance) and a resistor R₁​ in series.

(a) What is the minimum time that is needed to fully recharge the battery (with R₁​=0 Ω)

(b) What should the resistance R₁​ be to fully recharge the battery in exactly 1 hour?

Answer 1

a)

current in circuit with light bulb can be founded from light bulb has specs:

$I=P/U=1/14=0.071$ A

then, capacity of battery after 8 hours:

$2-0.071\cdot8=1.43$ Ah

for circuit with charger:

$\varepsilon=I_1r$

$I_1=\varepsilon/r=20/4=5$ A

so, the minimum time that is needed to fully recharge the battery:

$t=(2-1.43)/5=0.114$ hours $=6.84$ min

b)

$t=0.57/I_1=1$ hour

$I_1=\varepsilon/(r+R_1)=0.57$ A

$R_1=\varepsilon/I_1-r=20/0.57-4=31\ \Omega$