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Mobile App Chapter 6: Problem 4
A d.c. source has an electromotance of \(120 \mathrm{~V}\) and a negligible internal resistance. If \(n\) cells, each of electromotance \(2.1 \mathrm{~V}\) and internal resistance \(0.1 \Omega\), are to be charged from this source with a charging current of \(3 \mathrm{~A}\), find the series resistance necessary. If \(n=20\), what proportion of energy delivered by the source is wasted as heat? Find also the potential difference across the 20 cells.
Short Answer
Expert verified
Answer: The proportion of energy wasted as heat when 20 cells are charged by a 120 V DC source is approximately 24.17%, and the potential difference across the 20 cells is 80.0 V.
Step by step solution
01
Identify the given values
In this exercise, we are given:
Electromotance of the DC source (E) = 120 V
Number of cells, n
Electromotance of each cell = 2.1 V
Internal resistance of each cell (r) = 0.1 Ω
Charging current (I) = 3 A
02
Calculate total electromotive force of n cells
The total electromotive force of n cells is given by the product of the number of cells and the electromotance of each cell:
Total_EMF = n × 2.1 V
03
Calculate total internal resistance of n cells
The total internal resistance of n cells is given by the product of the number of cells and the internal resistance of each cell:
Total_Internal_Resistance = n × 0.1 Ω
04
Use Ohm's Law to find the necessary series resistance
According to Ohm's law: Voltage = Current × Resistance,
We will now replace the total EMF of the cells and internal resistance in this equation to find the necessary series resistance. Let's denote the series resistance as Rs:
120 V = 3 A × (Total_EMF + Total_Internal_Resistance + Rs)
Rs = (120 V - 3 A × (n × 2.1 V + n × 0.1 Ω)) / 3 A
05
Calculate the energy supplied and wasted when n=20
First, let's find the series resistance when n=20:
Rs = (120 V - 3 A × (20 × 2.1 V + 20 × 0.1 Ω)) / 3 A ≈ 0.333 Ω
Now, we can calculate the power supplied by the source as P_source = V × I:
P_source = 120 V × 3 A = 360 W
The power delivered to the cells is P_cells = (Total_EMF + Total_Internal_Resistance) × I
P_cells = (20 × (2.1 V + 0.1 Ω)) × 3 A ≈ 273 W
The power waste is the difference between the power supplied and delivered to the cells:
P_waste = P_source - P_cells = 360 W - 273 W = 87 W
To find the proportion of energy wasted, we can divide the wasted power by the power supplied:
Proportion of energy wasted = P_waste /P_source
Proportion ≈ 87 W / 360 W ≈ 0.2417 or 24.17%
06
Calculate the potential difference across the 20 cells
The potential difference across the 20 cells can be calculated using Ohm's law:
Potential_difference = Voltage - Current × (Total_Internal_Resistance + Rs)
Potential_difference = 120 V - 3 A × (20 × 0.1 Ω + 0.333 Ω) ≈ 80.0 V
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electromotive Force (EMF)
Electromotive force (EMF) is a term used to describe the energy provided by a power source, such as a battery or generator, per unit charge to drive electrons through a circuit. It is the force that pushes the electrons and creates current, and is measured in volts (V). However, it's important not to confuse EMF with potential difference, which is the voltage across two points in an external circuit. Even though both are measured in volts, EMF refers to the energy source while potential difference is about the energy that's available after losses like the energy lost as heat due to internal resistance.
Ohm's Law
Ohm's Law is a fundamental principle in electrical engineering and physics, describing the relationship between voltage (V), current (I), and resistance (R). It states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. Mathematically, this relationship is expressed as
\( V = I \times R \),
where V is the voltage in volts, I is the current in amperes (A), and R is the resistance in ohms (\(\Omega\)). Understanding this law is crucial when analyzing DC circuits, as it allows for the calculation of unknown quantities when the others are known.
\( V = I \times R \),
where V is the voltage in volts, I is the current in amperes (A), and R is the resistance in ohms (\(\Omega\)). Understanding this law is crucial when analyzing DC circuits, as it allows for the calculation of unknown quantities when the others are known.
Internal Resistance
Internal resistance refers to the inherent opposition to the flow of current within the power source itself, such as a battery. It acts like any other resistor in a circuit, causing energy to be wasted as heat when current flows. The internal resistance can affect the performance of the battery, reducing the available EMF as it translates to a lower terminal voltage when the current is drawn. Over time and with constant use, the internal resistance of a battery generally increases, leading to decreased efficiency.
Series Resistance
When we talk about series resistance in a circuit, we are referring to resistors arranged in a series connection, meaning they are aligned in a single path for the electric current to flow. The total resistance in a series circuit is just the sum of the individual resistances, which adds to the overall resistance in the circuit. This series resistance can be purposefully added, such as in charging circuits to limit the charging current to a safe value, preventing the circuit elements from being exposed to too much current.
Charging Current
Charging current is the current that flows through a circuit when charging a device like a battery. It's crucial to manage this current to ensure efficient and safe charging. If the current is too high, you may overheat the battery or even shorten its lifespan. In the context of our exercise, the charging current is limited to 3 A using a series resistance. This is done to prevent damage that can be caused by excessive current, ensuring the battery cells are charged at an optimal rate.
Energy Waste as Heat
Energy wastage in the form of heat within an electrical circuit is an unavoidable phenomenon due to resistance. When current passes through resistors, including the internal resistance of batteries or power sources, part of the electrical energy is converted into thermal energy. This waste of energy not only reduces the efficiency of the circuit but can also lead to overheating and potential damage. Therefore, it's essential to calculate the amount of power wasted as heat to design more efficient and safe circuits.
Potential Difference
Potential difference, often simply called voltage, is the measure of the work done to move a charge between two points in a circuit. It is the energy converted from electrical potential energy to another form of energy (like kinetic or thermal energy) per unit charge. In our exercise, the potential difference across the cells when they are being charged is less than the EMF of the charging source due to the voltage drop across the internal resistance and series resistance. It's this potential difference that is available for the cells to store the energy.
