3.11 - CELLS IN SERIES AND IN PARALLEL
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Understanding Series Cells
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Today, we're going to talk about how cells can be connected in series. Who can tell me what happens to the total voltage when cells are in series?
The total voltage increases!
Correct! When you connect cells in series, their voltages add up. If we have two cells with emfs of e₁ and e₂, the total emf, e_eq, becomes e₁ + e₂. Can anyone memorize that?
Uh, would it be 'E equals e₁ plus e₂'?
Exactly! You can think of it as 'Every cell adds energy!' Remember the acronym **E^2** for Double Energy. Now, what about the internal resistance?
Does it add up too?
Yes, it does! The equivalent internal resistance, r_eq, is equal to r₁ + r₂. So, more cells mean a higher total resistance. Can you see the importance of both total voltage and resistance?
Yes! If I have more cells in series, I have more voltage but also more resistance.
Right! In summary, when cells are in series, the equation for total voltage and internal resistance is crucial for understanding how to manage power in a circuit.
Understanding Parallel Cells
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Now, let's shift our focus to parallel connections of cells. What remains the same in a parallel combination?
The voltage stays the same!
Yes! In a parallel connection, the voltage across all cells is equal to the voltage of one of the cells. In other words, e_eq equals e₁ if all cells have identical emfs. But what happens to the current capacity?
The current increases, right?
Exactly! The total current is the sum of the currents from each cell. So if the cell currents are I₁ and I₂, the total current, I_eq, is I₁ + I₂. Would anyone like to remember this by creating another acronym?
Maybe we could use **C for Current**? So 'C equals I₁ plus I₂'?
Great suggestion! 【C = I₁ + I₂】 helps remember that combining cells in parallel increases the total current. However, how do we calculate the equivalent internal resistance?
Is it just like in series?
Not quite! It's actually the inverse: 1/r_eq = 1/r₁ + 1/r₂. This means that adding more cells decreases the overall resistance. Why is this beneficial?
It allows more current to flow without increasing the voltage!
Exactly! That’s why parallel arrangements are often used to achieve higher current capabilities while keeping the voltage consistent.
Real-world Applications of Series and Parallel Configurations
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Let's discuss where we might see series and parallel arrangements of cells in real life. Can anyone think of an example?
A flashlight uses batteries in series, right?
Correct! This setup gives it a stronger light because the voltage is higher. What about parallel arrangements?
Like in cars, where batteries can be in parallel to ensure the power supply can sustain more load?
Exactly! Cars often have multiple batteries in parallel to provide extra current when needed, like during starting. What do you think the disadvantage would be for a series connection?
If one battery dies, it won't work.
Right! In series, if one cell fails, the whole system fails. In contrast, in a parallel configuration, the failure of one cell does not affect the current supply from the other cells. That's the power of configuration.
Introduction & Overview
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Quick Overview
Standard
Understanding the arrangement of cells in series and parallel is vital for circuit design. Cells in series increase the total voltage while maintaining the same current, while parallel arrangements keep the voltage constant but increase the total current capacity. The internal resistances also combine differently in these configurations, affecting the overall circuit performance.
Detailed
Cells in Series and Parallel
In electric circuits, cells can be arranged in two main configurations: series and parallel. In a series configuration, the positive terminal of one cell is connected to the negative terminal of the next cell, resulting in the total electromotive force (emf) being the sum of the individual emfs of the cells. If the cells are denoted by their emfs as e₁ and e₂ and their internal resistances as r₁ and r₂, the combined emf and internal resistance of the configuration can be expressed as:
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Equivalent emf in series:
e_eq = e₁ + e₂ -
Equivalent internal resistance in series:
r_eq = r₁ + r₂
In contrast, when cells are arranged in parallel, each cell connects to the same two points in the circuit. This setup keeps the voltage the same as that of a single cell but increases the total available current capacity. For parallel configurations, the equations are:
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Equivalent emf in parallel:
e_eq = e₁ = e₂ (if they are identical) -
Equivalent internal resistance in parallel:
1/r_eq = 1/r₁ + 1/r₂
These configurations allow for enhanced flexibility in circuit design, leading to efficient utilization of battery power according to the requirements of various electrical devices.
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Introduction to Cells in Series
Chapter 1 of 3
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Chapter Content
Like resistors, cells can be combined together in an electric circuit. And like resistors, one can, for calculating currents and voltages in a circuit, replace a combination of cells by an equivalent cell.
Consider first two cells in series, where one terminal of the two cells is joined together leaving the other terminal in either cell free. Let e₁, e₂ be the emf’s of the two cells and r₁, r₂ their internal resistances, respectively. The potential difference between the terminals is also calculated based on the current flowing through the cells.
Detailed Explanation
When we connect two cells in series, we effectively combine their electromotive forces (emf) and internal resistances to create a single equivalent cell. The positive terminal of one cell connects to the negative terminal of the second, creating a continuous path for current to flow. The total emf of the series combination is simply the sum of their individual emf’s, while the total internal resistance adds up similarly. This leads to the formula:
- Total equivalent emf (e_eq) = e₁ + e₂
- Total equivalent resistance (r_eq) = r₁ + r₂
This means that by understanding the properties of cells in series, we can simplify complex circuits into simpler versions where calculations become easier.
Examples & Analogies
Think of cells in series like a group of people passing water buckets down a line to fill a large container. Each person represents a cell with their own ability to carry water (emf), and the water can only flow through one person at a time (internal resistance). The more people (cells) you add, the more water (current) you can ultimately deliver to the container.
Voltage and Current in Series Cells
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Chapter Content
The potential difference between the terminals A and C of the combination is:
V_AC = e₁ + e₂ - I(r₁ + r₂)
Detailed Explanation
The formula provided shows how to calculate the total potential difference (voltage) across terminals when two cells in series are connected with a load. Each cell's emf contributes positively to the total voltage, while the current flowing through the internal resistances (r₁ and r₂) creates a voltage drop, thereby reducing the effective voltage available across the load. Hence, to determine the voltage across the external circuit connected to the cells, you add the emf’s and then subtract the voltage drop caused by the internal resistances from the total current I.
Examples & Analogies
Imagine that each cell is like a fountain, where each fountain contributes its water flow (emf) to a main stream (total voltage). But there is a dirt path leading to the stream (internal resistance), which slows down the flow of water. The water that arrives at the stream is less than what the fountains could potentially provide because some is lost along the way—it’s the same with electricity.
Cells in Parallel Configuration
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Chapter Content
Next, consider a parallel combination of the cells. For connections across A and C, the currents I₁ and I₂ flow out of the positive electrodes, with the total current I flowing into the circuit. Since the potential difference across all cells in parallel is the same, we can write:
e_eq = e₁ = e₂
And for the internal resistances:
1/r_eq = 1/r₁ + 1/r₂
Detailed Explanation
In a parallel configuration, all the cells share the same voltage across their terminals, which means their emf's remain equal. Unlike in series, the currents from each cell contribute to a total current going through the circuit. The formula for calculating the equivalent internal resistance demonstrates how each cell's resistance decreases the overall resistance of the circuit. When cells are connected in parallel, the combined resistance is less than that of the lowest individual resistance.
Examples & Analogies
Think of parallel cells like several water hoses connected together at the same end. Each hose (cell) can deliver water independently into the same basin (circuit), which means that if one hose is blocked, the others can still deliver water. Because multiple supplies are combined, the overall water flow (current) increases, but since they share the same entrance (voltage), it's not doubled but instead divided among each pathway.
Key Concepts
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Series Connection: Increases total voltage but also total internal resistance.
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Parallel Connection: Maintains the same voltage while increasing total current capacity.
Examples & Applications
A flashlight typically uses batteries in series to increase the voltage for a brighter light.
Car batteries are often arranged in parallel to provide sufficient current for the starter motor.
Memory Aids
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Rhymes
In series they add, stacking up high, voltage is increasing, oh my oh my!
Stories
Imagine a string of pearls where each pearl represents a battery in series, creating a long, bright necklace of energy.
Memory Tools
For cell Sequence, remember V = V1 + V2 for Series, while for Parallel, I = I1 + I2.
Acronyms
In series, **S**ummate the voltage to get **S**trength.
Flash Cards
Glossary
- Electromotive Force (emf)
The energy provided per unit charge by a source of electrical energy, measured in volts.
- Internal Resistance
The resistance within a battery or cell that causes a drop in potential when current flows.
- Equivalent Resistance
The total resistance of a combination of resistors or cells in a circuit.
- Series Connection
A configuration where components are connected end-to-end, resulting in the same current flowing through each.
- Parallel Connection
A configuration where components are connected across common points, allowing for multiple paths for current.
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