Problem — AA battery for an LED torch

Problem

Estimate for how many hours an alkaline AA battery (EMF $1{,}5\ \text{V}$, charge $2500\ \text{mAh}$) can power a small LED torch that draws $20\ \text{mA}$. Compare with an incandescent torch that draws $\sim 250\ \text{mA}$.

Solution

Meaning of capacity — the charge $2500\ \text{mAh}$ is the product current $\times$ time that the battery can deliver: if a current $i$ flows, the battery lasts $t = \frac{Q}{i}$

LED torch ($i = 20\ \text{mA}$): $t_\text{LED} = \frac{2500\ \text{mAh}}{20\ \text{mA}} = 125\ \text{h}$

Incandescent torch ($i = 250\ \text{mA}$): $t_\text{inc} = \frac{2500\ \text{mAh}}{250\ \text{mA}} = 10\ \text{h}$

Comparison — the LED torch draws a current $12{,}5$ times smaller and lasts proportionally longer: $125\ \text{h}$ versus $10\ \text{h}$. At equal voltage the power is proportional to the current ($P = V\,i$), so the LED, much more efficient at converting electrical energy into light, consumes less and hugely extends battery life. (These are estimates: the real capacity depends on the discharge current and the cut-off voltage.)

$\ev{t_\text{LED} \sim 125\ \text{h} \quad (\text{vs} \sim 10\ \text{h for the incandescent torch})}$

Collegamenti

Argomenti: Circuiti elettrici Concetti: Forza elettromotrice Competenze: Stime di Fermi