Question

A \(11.45-V\) battery with internal resistance \(R_{i}=0.1373 \Omega\) is to be charged by a battery charger that is capable of delivering a current \(i=9.759 \mathrm{~A}\). What is the minimum emf the battery charger must be able to supply in order to charge the battery?

Short answer

Answer: The minimum emf the battery charger must supply to charge the battery is approximately 12.79 V.

Step-by-step solution

Identify the given values and their symbols

Given values: - Battery voltage, \(V = 11.45 V\) - Internal resistance, \(R_i = 0.1373 \Omega\) - Charging current, \(i = 9.759 A\) We need to find the minimum emf, \(E\), of the charger.

Apply Ohm's Law to find the voltage drop across the internal resistance

Ohm's Law states that \(V = iR\). In this case, we'll use it to find the voltage drop across the internal resistance (\(V_i\)) when the charging current is flowing through the battery. $$V_i = iR_i$$ Substitute the given values: $$V_i = (9.759 A)(0.1373 \Omega)$$ Calculate the voltage drop: $$V_i \approx 1.340 V$$

Find the minimum emf required to charge the battery

To charge the battery, the charger's emf must be higher than the sum of the battery's voltage and the voltage drop across the internal resistance. So, the minimum emf is: $$E_\text{min} = V + V_i$$ Substitute the values: $$E_\text{min} = 11.45 V + 1.340 V$$ Calculate the minimum emf: $$E_\text{min} \approx 12.79 V$$ So, the minimum emf the battery charger must be able to supply in order to charge the battery is approximately \(12.79 V\).