Question

The dead battery of your car provides a potential difference of \(9.950 \mathrm{~V}\) and has an internal resistance of \(1.100 \Omega\). You charge it by connecting it with jumper cables to the live battery of another car. The live battery provides a potential difference of \(12.00 \mathrm{~V}\) and has an internal resistance of \(0.0100 \Omega\), and the starter resistance is \(0.0700 \Omega\). a) Draw the circuit diagram for the connected batteries. b) Determine the current in the live battery, in the dead battery, and in the starter immediately after you closed the circuit.

Short answer

Answer: The currents in the live battery, dead battery, and the starter are 1.737 A each.

Step-by-step solution

Draw the circuit diagram

For drawing the circuit diagram, take the live battery, dead battery, and starter representing each with their respective voltage sources and the internal resistances. Connect the positive terminal of the live battery to the positive terminal of the dead battery, and the negative terminal of the live battery to the negative terminal of the dead battery. Place the starter in the middle of the positive jumper cable connecting the positive terminals of the batteries.

Determine the total resistance in the circuit

In order to find the total resistance in the circuit, we need to add the internal resistances of the live battery, dead battery, and the starter. The total resistance R_total can be calculated as: R_total = R_live + R_dead + R_starter = (0.0100 Ω) + (1.100 Ω) + (0.0700 Ω) = 1.180 Ω

Determine the total potential difference in the circuit

The potential difference across the live battery is given as 12.00 V, and across the dead battery is 9.950 V. The total potential difference across the circuit V_total can be calculated as the difference between the potential difference of the live and dead batteries: V_total = V_live - V_dead = (12.00 V) - (9.950 V) = 2.050 V

Determine the current in the circuit

Now that we have the total resistance and potential difference in the circuit, we can use Ohm's law to find the total current in the circuit. Ohm's law is given as: I_total = V_total / R_total I_total = (2.050 V) / (1.180 Ω) = 1.737 A (approx)

Determine the current in the live battery, dead battery, and the starter

Since the batteries and the starter are connected in series, the current flowing through them remains the same. Therefore, the current flowing in the live battery, dead battery, and the starter is 1.737 A. So, the currents in each component are as follows: I_live = I_dead = I_starter = 1.737 A

Key concepts

Ohm's Law

Understanding Ohm’s Law is crucial for anyone delving into circuit analysis. This fundamental principle relates the current flow through a conductor to the voltage across it and the resistance of the conductor. Formulated by Georg Simon Ohm in the 19th century, the law is elegantly simple:
\[ I = \frac{V}{R} \]
Here, \(I\) stands for the current in amperes (A), \(V\) is the potential difference in volts (V), and \(R\) represents the resistance in ohms (Ω). In essence, the law explains that the current is directly proportional to the voltage provided there is a constant resistance. This is clearly illustrated in step 4 of the exercise solution, where the current in the circuit is calculated by dividing the total potential difference by the total resistance. Ohm's law serves as the backbone for analyzing simple series and parallel circuits, and without it, we wouldn't be able to effectively determine the behavior of electrical components.

Electrical Resistance

Electrical resistance can be thought of as the 'friction' within the flow of current in an electrical circuit. It's a measure of the opposition to the movement of electrons through a conductor and is provided in ohms (Ω). Resistance depends on several factors including the material's nature, its temperature, length, and cross-sectional area.

• Materials with low resistance, such as copper and aluminum, are excellent conductors of electricity.
• Insulators like rubber or glass have very high resistance and discourage the flow of electrons.
• As the temperature of a conductor increases, typically so does its resistance.

For analyzing circuits, like the car battery charging example, the internal resistance of the batteries and the starter's resistance are summed up to find the total resistance that impacts the overall current flow (step 2 in the solution). It's important to understand how each component within a circuit contributes to the total resistance, as this will determine how much current will flow.

Current Flow

Current flow refers to the movement of electric charge through a conductive pathway, driven by a potential difference (voltage). It’s quantified in amperes, which are a measure of the charge flow per second. In a circuit, the flow of current can be direct (DC) or alternating (AC). For this car battery charging scenario, we're dealing with DC current.
Current will only flow through a circuit if it is provided a path, and for any amount of current flow to happen, there must be a potential difference. This potential difference motivates electrons to move from a point of high potential to low potential. In a series circuit, such as the one in the car battery problem, the same current flows through all components (step 5 in the solution). The current flowing through the live battery, dead battery, and the starter is the same in this scenario, demonstrating a fundamental rule of series circuits.

Potential Difference

Potential difference, also known as voltage, is the driving force behind the flow of current in a circuit. It’s the difference in electrical potential energy between two points in a circuit. Imagine it as akin to water pressure in a pipe—greater pressure means more water flow, just like greater voltage results in more current flow, assuming resistance is constant.
Voltage can be measured using a voltmeter, and in the given problem, we see the live battery has a potential difference of 12.00V, while the dead battery's potential difference stands at 9.950V (step 3 in the solution). The total potential difference that drives the current through the circuit is the difference between the voltages of these two batteries. Grasping this concept helps in understanding why electrical appliances require certain voltages to operate and why batteries need a particular voltage for charging.