A 12 V car battery dies not so much because its voltage drops but because chemical reactions increase its internal resistance. A good battery connected with jumper cables can both start the engine and recharge the dead battery. Consider the automotive circuit of FIGURE P28.64. How much current is the dead battery alone able to drive through the starter motor?

Verified step by step guidance

1. Identify the key components of the problem: The dead battery has an internal resistance (r_dead) and an electromotive force (emf_dead = 12 V). The starter motor has a resistance (R_motor). The goal is to calculate the current (I_dead) driven by the dead battery through the starter motor.
2. Apply Ohm's Law to the circuit formed by the dead battery and the starter motor. The total resistance in this circuit is the sum of the internal resistance of the dead battery (r_dead) and the resistance of the starter motor (R_motor). The total resistance is given by: $R{\mathrm{total}}_{}=r_{\mathrm{dead}}+R_{\mathrm{motor}}$.
3. The current driven by the dead battery can be calculated using Ohm's Law: $I{\mathrm{dead}}_{}=\frac{{\mathrm{emf}}_{\mathrm{dead}}}{R_{\mathrm{total}}}$. Substitute the values of ${\mathrm{emf}}_{\mathrm{dead}}$, $r_{\mathrm{dead}}$, and $R_{\mathrm{motor}}$ into this equation.
4. Simplify the equation to find the value of $I{\mathrm{dead}}_{}$. Ensure that the units are consistent (e.g., resistance in ohms, voltage in volts).
5. Interpret the result: The calculated current $I{\mathrm{dead}}_{}$ represents the amount of current the dead battery alone can drive through the starter motor. This value will likely be small due to the increased internal resistance of the dead battery.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as I = V/R. Understanding this law is crucial for analyzing how much current a battery can provide, especially when considering the internal resistance of the battery.

Internal Resistance

Internal resistance refers to the opposition to current flow within a battery itself, caused by the chemical processes occurring during discharge. As a battery discharges, its internal resistance can increase, reducing the effective voltage available to drive current through an external circuit. This concept is essential for determining how much current a dead battery can supply to a starter motor, as higher internal resistance leads to lower current output.

Battery Capacity and State of Charge

Battery capacity is a measure of the total charge a battery can store, typically expressed in ampere-hours (Ah). The state of charge indicates how much of that capacity is currently available for use. A dead battery may have a low state of charge, affecting its ability to deliver current. Understanding these concepts helps in evaluating the performance of a battery in starting an engine and the implications of its chemical reactions on current output.