# Electric Circuits. Series and Parallel Circuits. Short Circuits | iCalculator™

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15.4Electric Circuits. Series and Parallel Circuits. Short Circuits

In this Physics tutorial, you will learn:

• What is an electrical circuit?
• What are the symbols of electric circuit? Why do we use them?
• What are some of the circuit components? Which of them are necessary to make a circuit operate regularly?
• What are the sources of electricity? What is their operating principle?
• How do we combine the cells? What are the advantages and drawbacks each combination does provide?
• What kind of circuits are there? How do we calculate the quantities involved in these circuits?
• What are short circuits? What do they cause?
• How to protect electrical appliances from excessive current?

## Introduction

State the names of some electric appliances you know. What do they have in common? Where do they differ from each other?

Is it suitable to draw the circuit components in their original form? What is better to do in such cases?

What path does the current follow in an electrical circuit?

Which elements of a circuit do you think are necessary and which are optional? Explain.

This article is rich in content; it includes information from circuit elements to the methods of their combination in the circuit. In addition, some special cases of circuits encountered in everyday experience are discussed and explained.

## Electrical Circuits. Circuit Components

An electrical circuit is a closed-loop or path, which forms a network of electrical components, where electrons are able to flow. This path is made by conducting wires - which allow electrons to easily flow through them - and is powered by a source such as a battery.

An electrical circuit must necessarily include the following components, without which the circuit cannot operate:

• A power (electricity) source (batteries or plugs)
• Conducting wires which act as pathways for free electrons
• One or more electrical components that consume electricity

There are many other circuit components, which we will discuss in this paragraph.

It is not suitable to draw the shape of the circuit components because this takes too much time and not all people are good in drawing (similar to traffic symbols). Therefore, scientists have agreed to use simplified symbols in order to represent the circuit elements. As we have seen before, a straight line is used to represent a conducting wire, two parallel lines - one long and one short - are used to represent the cell, where the long line represents the positive terminal of the cell while the short line the negative one, a square is used to represent the resistor and so on. A simple circuit that includes the abovementioned components is shown below.

Some of the most frequently used circuit elements are shown in the table below:

Now, let's explore more in detail some of the above circuit components. In this way, you will know better their operating principle.

## Sources of Electricity

As stated earlier, in order to keep a steady flow of electrons through the circuit, we need a device that pushes the electrons and makes them flow continuously through the circuit. There are two major groups of devices, which can achieve this goal. The first group consists on batteries that produce one-directional current or direct current, (DC) and the other group consists on electric generators that produce alternating current (AC), i.e. a kind of current which charges its direction continuously (60 times per second in most countries of Americas and 50 times per second in the rest of the world). Here, we will focus on the DC current because AC current will be discussed in the next chapter.

As explained earlier, a cell is the basic element of a battery. Therefore, in this article we will explain the operating principle of various types of cells only. Basically, a cell converts chemical energy into electricity. There are three main types of cells:

### a. Wet (voltaic) cells

A wet cell is made up by a copper and a zinc plate immersed inside a weak acid solution. The acid solution is called electrolyte and it is used to conduct electricity, while the plates are called electrodes. The free electrons emerge from the zinc plate (anode) and move to the copper plate (cathode) when a conducting wire connects the two electrodes. As a result, an electric current is produced.

This type of cell can produce a 1V potential difference. Here, the carriers of electricity inside the electrolyte are positive and negative ions while electrons are carriers of electricity in the rest of the circuit. The electrolyte "scours" the electrodes and makes positive ions move from zinc to copper electrode while negative ions move in the opposite direction. As a result, the zinc plate is charged positively and copper plate negatively. The flow of electrons through the conducting wire tends to set up the balance by transferring electrons from zinc to copper. However, the scouring effect of electrolyte meanwhile continues, resulting in new ions produced. This process lasts for a long time, producing as a result a steady current in the circuit.

Wet cells are the oldest type of cells. The first wet cell was invented by Alessandro Volta, an Italian scientist who used salt-soaked leather as an electrolyte. Therefore, the unit of potential difference bears his name (Volt) in honour to his work.

### b. Dry cells

Wet cells have many limitations, from the low potential difference they produce (1V) to the difficulty in transportation due to the acidic liquid contained in them. Therefore, during their attempts to improve the battery, scientists invented another type of cell - the dry cell.

A dry cell consists of a zinc cylindrical can (anode) and a carbon rod at the centre (cathode). A chemical paste here substitutes the effect of acidic solution used in the wet cells.

The operating principle of dry cells is the same as in wet cells. Dry cells produce a potential difference of 1.5 V (50% more than wet cells); furthermore, they are easy to transport as you can rotate them without problem.

### c. Accumulators

Accumulators are also known as "storage batteries." They are used in cars, laptops, flashlights etc. Accumulators are groups of wet cells enclosed inside plastic containers. Zinc and copper electrodes here are replaced by lead electrodes. Each cell of accumulators produce 2V of potential difference. This is the reason why they are preferable when a higher voltage is needed.

There are 6 cells accumulators (in total they produce 12 V) and 9 cells accumulators (in total they produce 18 V). In addition, the electrolyte of accumulators is made up from dilute sulphuric acidic solution.

Accumulators have an important advantage: they are all rechargeable.

## Combination of Cells

Cells - just like resistors - can be combined in series and in parallel. Such combinations are made for practical purposes or for necessity.

### a. Series combination of cells

When the positive terminal of one cell is placed in contact with the negative terminal of another cell, we say they are connected in series, as shown in the figure below.

A series combination of cells offers the advantage of the increase in electromotive force as

emftotal=emf1 + emf2

Thus for two dry cells connected in series, we obtain

emftotal = 1.5 V + 1.5 V
= 3 V

However, there is a drawback when connecting two cells is series: their life is not so long as they have to work at full capacity.

### b. Parallel combination of cells

When two cells are placed side by side in a circuit, we say they are connected in parallel as shown in the figure below.

A parallel combination of cells has an advantage to the series combination: the batteries last longer as they help each other to do the required work. This results in a slower consumption of each battery.

However, a parallel setup of cells has a serious drawback: the potential difference is the same as if there was a single cell in the circuit. This means the electric source shown in the above figure produces an electromotive force of 1.5 V, as it is composed by two dry cells.

## Series and Parallel Circuits

Now, let's extend the concept of series and parallel combination of resistors to include the other quantities such as potential difference and current as well. Before continuing with numerical examples, first let's prove the correctness of the two formulae used in the combination of resistors which we have provided in article 14.2.

Let's consider two resistors connected in series in a circuit, as shown in the figure below.

Since there is the same current I flowing through both resistors, we can write

Itot = I1 = I2 = I

Also, we have

emf = ∆V1 + ∆V2

Applying Ohm's Law, in the last equation, we obtain

I ∙ Req = I ∙ R1 + I ∙ R2

Simplifying the current I from both sides, we obtain the known formula of series combination of resistors

Req = R1 + R2

### Example 1

What is the value of resistance R2 in the circuit shown below? Neglect the resistance of conductor and that of battery.

### Solution 1

We have the following clues:

emf = 24V
I = 3A
R1 = 7Ω
R2 = ?

First, we calculate the total (equivalent) resistance using Ohm's Law formula for the whole circuit. We have:

Req = emf/I
= 24V/3A
= 8Ω

Since the resistors are connected in series, we have:

Req = R1 + R2
R2 = Req - R1
= 8Ω - 7Ω
= 1Ω

Now, let's consider two resistors connected in parallel, as shown in the figure below.

The current I divides in the two branches, and the corresponding values of current in each branch are I1 and I2 respectively. The potential difference in each resistor is the same as that in the entire branch. If the resistor of wire and that of battery are neglected, we obtain:

emf = ∆V1 = ∆V2 = ∆V

Also, the equation for currents is

I = I1 + I2

Applying Ohm's Law in the last equation, we obtain

emf/Req = ∆V1/R1 + ∆V2/R2
∆V/Req =∆V/R1 + ∆V/R2

Simplifying ΔV from both sides, we obtain for parallel combination of resistors the known equation:

1/Req =1/R1 + 1/R2

### Example 2

What is the current in the main branch and the current flowing in each parallel branch of the circuit shown below? Use two methods for solving this exercise.

### Solution 2

Method 1

This method consist in calculating the equivalent resistance, which helps finding the current in the main branch first. Then, since potential difference in each branch of the parallel setup is considered as equal the electromotive force of battery, we uses Ohm's Law to calculate the current in each resistor.

The equivalent resistance of the (parallel) circuit is:

1/Req = 1/R1 + 1/R2
= 1/6 + 1/12
= 2/12 + 1/12
= 3/12

Thus,

Req = 12/3 Ω
=4Ω

Now, let's calculate the current I in the main branch using Ohm's Law. We have

I = emf/Req
= 18 V/4 Ω
= 4.5 A

Since the electromotive force of battery corresponds to the potential difference in each wire of the parallel branch, we obtain for the currents I1 and I2 flowing in these wires:

I1 = emf/R1
= 18 V/6 Ω
= 3 A

and

I2 = emf/R1
= 18 V/12 Ω
= 1.5 A

Method 2

This method consists in finding the currents in each wire directly by applying Ohm's Law. Then, we add these currents to find the current in the main branch. Thus, we have:

I1 = emf/R1
= 18 V/6 Ω
= 3A

and

I2 = emf/R2
=18 V/12 Ω
= 1.5A

Therefore, the current in the main branch is

I = I1 + I2
= 3A + 1.5A
= 4.5A

As you can see, the results are the same in both cases.

## Short Circuits

It is a known fact that the electric current chooses the easiest path to flow. This path is not meant to be the shortest one. When we bypass a resistor through an extra conducting wire added in that part of the circuit, this action brings the formation of a short circuit, which is a circuit with a very low resistance (just the resistance of the conductor and that of the source), and which brings an increase of the current in the circuit. In other words, a short circuit "cancels" the effect of the resistance it bypasses. Look at the figure.

### Example 3

An electric circuit composed by three resistors R1 = 18 Ω, R2 = 12 Ω and R3 = 9 Ω connected in parallel to a 24 V battery is shown in the figure below.

The resistance of wire is Rw = 0.5 Ω and that of battery is r = 1 Ω.

Calculate the current in the main branch if:

1. the circuit is in the actual state
2. the first resistor cuts in two pieces which separate from each other leaving a gap
3. An additional piece of wire is used to bypass the first resistor as shown in the figure in theory.

### Solution 3

a. First, we have to calculate the equivalent resistance of the parallel branch. We have,

1/Req = 1/R1 + 1/R2 + 1/R3
= 1/18 + 1/12 + 1/9
= 2 + 3 + 4/36
= 9/36

Thus,

Req = 36/9 Ω
= 4Ω

The total resistance in the circuit is

Rtot = Req + Rw + r
= 4Ω + 0.5Ω + 1Ω
= 5.5Ω

Therefore, the current in the main branch is

I = emf/Rtot
= 24 V/5.5 Ω
= 4.36 A

b. If the first resistor is faulty and cuts in two pieces, no current flows through it. As a result, only the other two resistors are in use. We use the same procedure as in part (a) to calculate the current in the main branch, i.e.

1/Req = 1/R2 + 1/R3
= 1/12 + 1/9
= 3 + 4/36
= 7/36

Thus,

Req=36/7 Ω
=5.14 Ω

The total resistance in the circuit is

Rtot = Req + Rw + r
= 5.14 Ω + 0.5Ω + 1Ω
= 6.64Ω

Therefore, the current in the main branch is

I = emf/Rtot
= 24 V/6.64 Ω
= 3.61 A

c) If we connect a piece of wire to bypass the first resistor, all the other resistors are bypassed as well. Therefore, only the resistance of wire and battery are to be considered. We have

Rtot = Rw + r
= 0.5Ω + 1Ω
= 1.5Ω

Therefore, the current in the main branch is

I = emf/Rtot
= 24 V/1.5 Ω
= 16 A

As you see, when a short circuit does occur, the current in the main branch is much higher than usual. Therefore, we must avoid such situations as a short circuit may cause damage to electrical appliances. To protect the appliances from such unexpected situations, producers usually install some inexpensive devices called fuses, which are connected in series with them. A good fuse holds a maximum current up to 0.5 A higher than the operating current of the appliance. Thus, for example, if an appliance operates at 8A current, a fuse with a maximum operating capacity of 8.5 A is installed before the appliance. In cases of short circuits when the current in the circuit becomes much higher than the operating current of the appliance, the fuse connected in series to it breaks first, protecting in this way the corresponding appliance as the current flow stops.

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