# What will be the minimum time to fully recharge this battery?

There is a rechargeable battery of EMF $$ε$$, internal resistance $$r$$ and battery capacity (after full charge) $$b$$ (Ah).

The fully charged battery is connected to a light bulb for $$t$$ hours with specs $$k$$ volts, and $$p$$ watts. Assume that the light bulb has constant resistance (independent of the current) and the EMF of the battery remains constant.

After $$t$$ hours, this battery is connected to a charging circuit with an ideal voltage source with EMF $$E$$, and a resistor $$R_1$$ in series.

The question is what is the minimum time to fully recharge this battery?

I have some questions (apart from the above) of myself which am unable to understand:

1. What does the "light bulb has constant resistance (independent of the current)" mean in this context?

2. How does the battery capacity in Ah help in the question?

I thought that the total energy from the battery will be $$ε.b$$ in Wh.

The power used up by the internal resistance will be $$P_r = ε^2r/(r+R)^2$$ and the power used by the bulb will be $$P_R = ε^2R/(r+R)^2$$.

I am not sure if $$ε.b = P_r.t + P_R.t$$ be a correct equation.

I would appreciate any help!

Edit:

Still a bit confused regarding the solution I came up with.

The total energy remaining after being connected to the circuit for $$t$$ hours will be

$$ε.b - p.t - \frac{ε^2rt}{r+R} - \frac{ε^2Rt}{r+R}$$

Which becomes $$ε.b - p.t - \frac{ε^2t}{r+R}$$

Now the question is to find the minimum time to fully recharge this battery.

This means the new EMF source must provide the lost energy to the old battery.

$$\frac{(E-ε)^2t_{min}}{r+R_1} = p.t + \frac{ε^2t}{r+R}$$

Is this is a correct solution? I also am not sure why this would be the minimum time.

What does the "light bulb has constant resistance (independent of the current)" mean in this context?

This just provides you with a simplifying assumption. The resistance of a real (incandescent) light bulb would change depending on the temperature of the filament and therefore the current through it, which might make the problem more difficult.

How does the battery capacity in Ah help in the question?

I'm not sure it is necessary information.

As you work towards a solution, I would recommend considering Energy instead of Power.

Remember to convert all the times to seconds, and remember the energy lost (to heat) in the internal resistance of the battery when both discharging and recharging.

• @PhilipWood my apologies - you are quite right. Jan 13, 2022 at 14:43
• Hey, do you think the solution I added above will be correct. I am not sure why this would be the minimum time (if it is correct anyway). What do you think? Jan 13, 2022 at 16:01