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(a) Find the terminal voltage of a 12.0-V motorcycle battery having a \(0.600-\Omega\) internal resistance, if it is being charged by a current of \(10.0 \mathrm{~A}\). (b) What is the output voltage of the battery charger?

Short Answer

Expert verified
The terminal voltage is 18.0 V, and the charger's output is also 18.0 V.

Step by step solution

01

Understanding the Problem

We need to find the terminal voltage of a 12.0 V battery with an internal resistance of 0.600 Ω when charged with a 10.0 A current. This will also allow us to find the charger’s output voltage.
02

Formula for Terminal Voltage

Terminal voltage (V_terminal) of a battery being charged is given by the formula: \( V_{\text{terminal}} = V_{\text{emf}} + I \times r \), where \( V_{\text{emf}} \) is the electromotive force (12.0 V here), \( I \) is the current (10.0 A), and \( r \) is the internal resistance (0.600 Ω).
03

Calculate Terminal Voltage

Substitute the values into the formula: \( V_{\text{terminal}} = 12.0 + (10.0 \times 0.600) \). This simplifies to \( V_{\text{terminal}} = 12.0 + 6.0 = 18.0 \) V.
04

Charging Output Voltage

During charging, the battery charger must provide voltage equal to the battery's terminal voltage. So in this case, the charger's output voltage is 18.0 V as it must overcome both the battery's EMF and the voltage across the internal resistance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Battery Voltage
Battery voltage is a fundamental concept in electric circuits, encompassing the electromotive force (emf) produced by the battery. Think of it as the driving force that pushes electrons throughout the circuit. This force is usually measured in volts (V). The more voltage a battery has, the more energy it can deliver per charge to the circuit. When we refer to battery voltage in a practical scenario, it's essential to distinguish between the open-circuit voltage and the terminal voltage. The open-circuit voltage refers to the battery's voltage when no current flows through it, essentially the emf. For a 12.0 V motorcycle battery, this is what we expect when measuring across the terminals in a no-load condition. However, during operation—specifically when the battery is connected to a load or charging system—the voltage at the terminals shifts. Influenced by internal resistance and the current flowing, this becomes the terminal voltage. Understanding this difference is crucial for diagnosing battery health and performance.
Internal Resistance
Internal resistance is an intrinsic property of all batteries, which opposes the flow of electric current. It behaves like a small resistor within the battery and is measured in ohms (Ω). This resistance causes energy losses in the form of heat, slightly reducing the available power from the battery. When dealing with circuits, we need to precisely consider internal resistance to calculate the effective voltage delivered by the battery. For instance, if a 0.600 Ω internal resistance exists in a 12.0 V battery and a current of 10.0 A flows through, a portion of the battery's voltage is "lost" across this resistance. The equation we use is: - Voltage Lost = Current × Internal Resistance = 10.0 A × 0.600 Ω = 6.0 V. This decrease means the battery's terminal voltage is increased by 6.0 V when charging due to the internal resistance adding to the emf.
Charging Current
Charging current refers to the electrical current supplied to a battery from a charger. The unit of measurement here is amperes (A). This current is vital as it dictates how quickly a battery gets charged. However, it's also vital in determining the terminal voltage. When a battery is being charged, understanding the interplay between the charging current and the battery's internal resistance is crucial. High currents can lead to significant voltage changes across the internal resistance, which in turn affect the terminal voltage. In the case of the 12.0 V motorcycle battery with a 10.0 A charging current, the internal resistance's influence is clear. The terminal voltage becomes 18.0 V, having increased by: - Voltage via charging = emf + (Charging Current × Internal Resistance) = 12.0 V + 6.0 V. Properly managing charging current is essential for maintaining battery health and ensuring efficient charging.

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