Exercise EE1-03.R1 (Solution)

Prof. Dr. Thorsten Jungmann (last updated 2025-12-07)

Relates to EE1-03 Sources

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Task

A portable LED torch is powered by an AA alkaline battery. The battery has a rated voltage of $U_0=1.5\mathrm{V}$ and a battery capacity of $Q=2500\mathrm{mAh}$.

In operation, the light emitting diode (LED) circuit draws a constant current of $I=200\mathrm{mA}$.

a) In theory, how long can the torch be operated before the battery is exhausted?

b) How much energy (in joules and watt-hours) is stored in the battery?

c) What power does the LED circuit consume?

Solution

a) $t=12.5\mathrm{h}$
b) $W=13500\mathrm{J}=3.75\mathrm{Wh}$
c) $P=0.3\mathrm{W}$

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Solution

Given

Given explicitly:

• Rated voltage of the battery: $U_0=1.5\mathrm{V}$
• Battery capacity: $Q=2500\mathrm{mAh}$
• Current draw of the LED circuit: $I=200\mathrm{mA}$

Known:

• Battery capacity = amount of charge that can be drawn $Q=I\cdot t$
• Energy: $W=P\cdot t=U\cdot I\cdot t=U\cdot Q$
• Power: $P=U\cdot I$

where:

• $Q$ is the electric charge in ampere-hours ($\mathrm{Ah}$) or coulombs ($\mathrm{C}$)
• $I$ is the current in amperes ($\mathrm{A}$)
• $t$ is the time in hours ($\mathrm{h}$) or seconds ($\mathrm{s}$)
• $W$ is the energy in joules ($\mathrm{J}$) or watt-hours ($\mathrm{Wh}$)
• $P$ is the power in watts ($\mathrm{W}$)
• $U$ is the voltage in volts ($\mathrm{V}$)

Required

a) Operating time $t$ in hours
b) Stored energy $W$ in $\mathrm{J}$ and $\mathrm{Wh}$
c) Power consumed $P$ in $\mathrm{W}$

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a) Operating time

The operating time is the battery capacity divided by the discharge current:

$$t=\frac{Q}{I}=\frac{2500\mathrm{mAh}}{200\mathrm{mA}}=\underline{12.5\mathrm{h}}$$

Note

This is a theoretical estimate. In practice, the battery voltage drops towards the end of the discharge, so the usable operating time may be shorter. In addition, the actual battery capacity depends on the discharge current – at higher currents, the effective battery capacity is lower.

b) Stored energy

The energy in watt-hours:

$$W=U_0\cdot Q=1.5\mathrm{V}\cdot 2500\mathrm{mAh}=1.5\mathrm{V}\cdot 2.5\mathrm{Ah}=\underline{3.75\mathrm{Wh}}$$

Conversion to joules (with $1\mathrm{Wh}=3600\mathrm{J}$):

$$W=3.75\mathrm{Wh}\cdot 3600\frac{\mathrm{J}}{\mathrm{Wh}}=\underline{13500\mathrm{J}}$$

Check via $W=P\cdot t$: with $P=0.3\mathrm{W}$ (see c) and $t=12.5\mathrm{h}$:

$$W=0.3\mathrm{W}\cdot 12.5\mathrm{h}=3.75\mathrm{Wh}$$

✔

c) Power consumed

The power consumed by the LED circuit:

$$P=U_0\cdot I=1.5\mathrm{V}\cdot 200\mathrm{mA}=1.5\mathrm{V}\cdot 0.2\mathrm{A}=\underline{0.3\mathrm{W}}$$

Typical values

A power consumption of $0.3\mathrm{W}$ is realistic for a small LED torch. For comparison: a classic torch with an incandescent bulb of comparable brightness would need about $1$ to $3\mathrm{W}$ and would discharge the battery correspondingly faster.