Exercise EE1-02.01 – Electric charge, voltage and energy of a rechargeable battery (Solution)

Prof. Dr. Thorsten Jungmann (last updated 2025-10-21)

Relates to EE1-02 Electrical quantities and units, terms, symbols and models

Task

A rechargeable battery (‘battery’ for short) carries the following manufacturer specifications: AA (Mignon), NiMH, 1 600 mAh, voltage 1.2 V.

a) What electric charge $Q$ does the fully charged rechargeable battery store?

b) What is the theoretical charging time of the rechargeable battery at a charging current of $I=400\mathrm{mA}$?

c) At a discharge current of $I=80\mathrm{mA}$, how long does it take until the rechargeable battery has discharged by $70\%$?

d) How many electrons are then still stored in the rechargeable battery?

e) What mechanical work could a remote-controlled car with an efficiency of $50\%$ perform if it is supplied by a rechargeable battery pack with a rated voltage of $4\cdot 1.2\mathrm{V}=4.8\mathrm{V}$ and four fully charged rechargeable batteries of the type mentioned above, and if all the stored energy were delivered?

◀️ Back to the task

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Solution

Given:

• Battery capacity $C=1600\mathrm{mAh}=1.6\mathrm{Ah}$
• Cell voltage $U=1.2\mathrm{V}$

Known:

• Elementary charge $e=1.602\cdot 10^{-19}\mathrm{C}$

a) Stored electric charge $Q$

$$Q=C\cdot 3600s/h=1.6\mathrm{Ah}\cdot 3600s/h=\underline{5760\mathrm{C}}$$

The fully charged rechargeable battery stores an electric charge of $5760\mathrm{C}$.

b) Theoretical charging time

$$t=\frac{Q}{I}=\frac{5760\mathrm{C}}{400\mathrm{mA}}=\frac{5760\mathrm{A}\mathrm{s}}{0.4\mathrm{A}}=14400\mathrm{s}=\underline{4\mathrm{h}}$$

Charging takes 4 hours.

c) Discharge time for 70 % discharge

Electric charge sought: $Q_{70}=0.7\cdot 5760\mathrm{C}=4032\mathrm{C}=4032\mathrm{A}\mathrm{s}$.

$$t=\frac{Q_{70}}{I}=\frac{4032\mathrm{A}\mathrm{s}}{80\cdot 10^{-3}\mathrm{A}}=50400\mathrm{s}=14\mathrm{h}$$

Discharging from $100\%$ to $30\%$ remaining charge takes 14 hours.

Alternatively, assuming that charging and discharging follow the same time behaviour, the time could also be determined from the result of b) as follows:

$$t_{70\%,80mA}=t_{100\%,400mA}\cdot \frac{400\mathrm{mA}}{80\mathrm{mA}}\cdot 0.7=4\mathrm{h}\cdot 5\cdot 0.7=\underline{14\mathrm{h}}$$

This would be imprecise, but in this case not wrong. The calculation above is, however, more reliable.

d) Remaining electrons

Remaining electric charge: $Q_{30}=0.3\cdot 5760\mathrm{C}=1728\mathrm{C}$.

$$n=\frac{Q_{30}}{e}=\frac{1728\mathrm{C}}{1.602\cdot 10^{-19}\mathrm{C}}\approx 1.08\cdot 10^{22}\text{electrons}$$

With a remaining charge of $30\%$, approximately 10 sextillion electrons remain in the rechargeable battery.

e) Mechanical work

Rechargeable battery pack: 4 cells in series → $U_{\mathrm{pack}}=4\cdot 1.2\mathrm{V}=4.8\mathrm{V}$.
Stored energy:

$$W_{\mathrm{battery}}=Q\cdot U_{\mathrm{pack}}=5760\mathrm{C}\cdot 4.8\mathrm{V}=27648\mathrm{J}$$

Mechanical work:

$$W_{\mathrm{mech}}=\eta \cdot W_{\mathrm{battery}}=0.5\cdot 27648\mathrm{J}=13824\mathrm{J}\approx \underline{13.8\mathrm{kJ}}$$

The device can perform $13.8\mathrm{kJ}$ of work.

Energy density of rechargeable batteries

With the rechargeable battery pack, a remote-controlled car weighing 1 kg could travel just under 15 km on a level track at a velocity of 20 km/h.
This is the energy of 1–2 gummy bears or half a millilitre of petrol. The comparison shows the lower energy density of rechargeable batteries compared with other energy carriers. This is why the rechargeable battery in an electric car takes up considerably more volume than a petrol tank, while providing a shorter range.