Combination of resistors

RESISTANCE OF A COMBINATION OF RESISTORS 

There are two ways of combining the resistors in a circuit. Figure below shows an electric circuit in which three resistors having resistances R1, R2 and R3, respectively, are joined end to end. Here the resistors are said to be connected in series. In series, there is only one path for flow of current.



Next Figure shows a combination of resistors in which three resistors are connected together between points X and Y. Here, the resistors are said to be connected in parallel. In parallel, there is separate path for flow of current in each resistor.

Resistors in Series 

In a series combination of resistors 
(1) Same current I flow through each resistor. 
(2) Potential difference across each resistor is different. V1 across R1, V2 across R2 and V3 across R3. 
(3) Total potential difference across the combination is equal to the sum of potential difference across the individual resistors. That is,
\[V=V_1+V_2+V_3\]

Equivalent Resistance of Series Combination

Let I be the current through the circuit. The current through each resistor is also I. Applying the Ohm’s law to three resistors separately, we have
\[V_1=IR_1\]
\[V_2=IR_2\]
\[V_3=IR_3\]

Since
\[V=V_1+V_2+V_3\]

We have
\[V=IR_1+IR_2+IR_3\]
\[V=I(R_1+R_2+R_3)\]
\[\frac{V}{I}=R_1+R_2+R_3\]
or
\[R_s=R_1+R_2+R_3\]


Where, RS is the equivalent resistance of the series combination. 

We can conclude that when several resistors are joined in series, the resistance of the combination RS equals the sum of their individual resistances, R1, R2, R3, and is thus greater than any individual resistance. 

We can imagine a single resistor RS replacing the three resistors joined in series such that the potential difference V across it, and the current I through the circuit remains the same.

Resistors in Parallel 

In parallel combination of resistors 

(1) Potential difference across each resistor is same. 
(2) Current through each resistor is different. I1 across R1, I2 across R2 and I3 across R3. 
(3) The total current I, is equal to the sum of the 
separate currents through each resistor of the 
combination.
\[I=I_1+I_2+I_3\]

Equivalent resistance in parallel combination

On applying Ohm’s law to each resistor of parallel combination, we have
\[I_1=\frac{V}{R_1}\]
\[I_2=\frac{V}{R_2}\]
\[I_3=\frac{V}{R_3}\]
Since
\[I=I_1+I_2+I_3\]

We have
\[I=\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}\]

\[I=V(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3})\]
\[\frac{I}{V}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\]
\[\frac{1}{R_P}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\]

Thus, we may conclude that the reciprocal of the equivalent resistance of a group of resistances joined in parallel is equal to the sum of the reciprocals of the individual resistances.

Preference of parallel combination over series 

We prefer parallel combination over series combination in domestic circuit because of the following reasons: 

(1) In a series circuit the current is constant throughout the electric circuit. Thus it is impracticable to connect different appliances such as an electric bulb and an electric heater in series, because they need currents of different values to function properly. 
(2) Another major disadvantage of a series circuit is that when one electrical device fails the circuit is broken and none of the devices connected in the circuit works.

On the other hand in parallel circuit...

(1) In a parallel circuit, different appliances are connected in different branches and each appliance gets its required amount of current in that branch. 
(2) In a parallel circuit, if one component fails, the others are not affected.

NOW CHECK YOUR PROGRESS!!! 

1. A wire of resistance R is cut into five equal parts. These five parts are then connected in parallel. If the equivalent resistance of this combination is R', then calculate the ratio R/R'. 

2. What is the (a) highest and (b) lowest, total resistance that can be obtained by combining four resistors of values 4Ω, 8Ω, 12Ω and 24Ω? 

3. How can three resistors of resistances 2Ω, 3Ω and 6Ω be connected to give a total resistance of (a) 4Ω (b) 1Ω ? 

4. Two resistors with resistances 5Ω and 10Ω respectively are to be connected to a battery of 6V so as to obtain (i) minimum current flowing (ii) maximum current flowing 

(a) How would you connect the resistance in each case? 
(b) Calculate the strength of the total current in the circuit in the two cases. 

5. A battery of 9V is applied across resistors of 0.2Ω, 0.3Ω, 0.4Ω, 0.5Ω and 12Ω connected in series. How much current would flow through the 12Ω resistors? 

6. How many 176 Ω resistors (in parallel) are required to carry 5 A on a 220 V line? 

7. A hot plate of an electric oven connected to a 220 V line has two resistance coils A and B, each of 24 Ω resistance, which may be used separately, in series, or in parallel. What are the currents in the three cases? 

8. In the given figure R1 = 10 Ω, R2 = 40 Ω, R3 = 30 Ω, R4 = 20 Ω, R5 = 60 Ω, and a 12 V battery is connected to the arrangement.

 Calculate (a) the total resistance in the circuit, and (b) the total current flowing in the circuit. 

9. An electric lamp, whose resistance is 20 Ω, and a conductor of 4 Ω resistance are connected in series to a 6 V battery. Calculate (a) the total resistance of the circuit, (b) the current through the circuit, and (c) the potential difference across the electric lamp and conductor. 

10. An electric lamp of 100 Ω, a toaster of resistance 50 Ω, and a water filter of resistance 500 Ω are connected in parallel to a 220 V source. What is the resistance of an electric iron connected to the same source that takes as much current as all three appliances, and what is the current through it? 

11. Three resistors of resistances 5Ω, 10Ω and 30Ω are connected in parallel across a 12V battery. Calculate: 
(a) Total resistance in the circuit. 
(b) Total current in the circuit. 
(c) Current through each resistor. 

Resistance and resistivity

Resistance: 

The flow of electrons is not so free inside a metallic conductor. Just like  flow of water in a river is opposed by the presence of big rocks in its way, the flow of electrons is similarly opposed by stationary atoms and repulsion by other electrons.

The typical speed of an electron moving inside a metal is of the order of \(10^6\) m/s. It however collides million times in a second and it barely moves 1mm in a unit second.

This property of a conductor to resist the flow of charge through it is called resistance.

SI unit :

Its SI unit is ohm, represented by the Greek letter Ω.

Definition of 1Ω: 

If the potential difference across the two ends of a conductor is 1 V and the current through it is 1 A, then the resistance R, of the conductor is 1 Ω. That is,
\[R=\frac{V}{I}\]
\[1\;ohm=\frac{1\;volt}{1\;ampere}\]
\[\textrm{or }1\;Ω=\frac{1\;V}{1\;A}\]

Relation between current and resistance: 

According to Ohm’s Law, 

\[I=\frac{V}{R}\]

the current through a resistor is inversely proportional to its resistance. If the resistance is doubled the current gets halved.

Since I is inversely proportional to the resistance of the circuit for a given V, we use this concept in controlling the amount of current in a domestic circuit by using a variable resistors.

Application of varying resistance: 

In many cases it is necessary to increase or decrease the current in an electric circuit like in case of an electric fan regulator and in electric iron for changing the amount of heat produced. The device used to control current without changing the voltage is called variable resistance.

 Symbol of variable Resistance:

Symbol of Rheostat ( a variable resistor):

Resistance of different material is different: 

The motion of electrons in an electric circuit constitutes an electric current. The electrons, however, are not completely free to move within a conductor. They are restrained by the attraction of the atoms among which they move. Thus, motion of electrons through a conductor is retarded by its resistance.

• A component of a given size that offers a low resistance is a good conductor.
• A conductor having some appreciable resistance is called a resistor.
• A component of identical size that offers a higher resistance is a poor conductor.
• An insulator of the same size offers even higher resistance.

Factors on which the resistance of a conductor depends: 

Resistance of the conductor is 

(i) Directly proportional to its length, 

(ii) Inversely proportional to its area of cross-section, and 

(iii) On the nature of its material. 

That is, 

\[R∝l ----(1)\]

\[R∝\frac{1}{A}----(2)\]

Combining (1) and (2)

\[R∝\frac{l}{A}\]

Or

\[R∝ρ\frac{l}{A}\]

Where ρ (rho) is a constant of proportionality and is called the electrical resistivity of the material of the conductor. 

Resistivity of a given material is independent of the dimension (i.e. length and area of cross-section) of given material and it depends only on nature of material and temperature. 

SI unit of resistivity: 

The SI unit of resistivity is Ω m. It is a characteristic property of the material.

Remarks: 

(1) The metals and alloys have very low resistivity in the range of 

\[10^{-8}ᘯ\;m\quad\textrm{to}\quad 10^{-6}ᘯ\;m\]

They are good conductors of electricity.

(2) Insulators like rubber and glass have resistivity of the order of

\[10^{12}ᘯ\;m\quad\textrm{to}\quad 10^{17}ᘯ\;m\]

(3) Both the resistance and resistivity of a material vary with temperature.

(4) Resistivity of a material does not change on changing the length and area of cross-section of the conductor.

(5) Resistivity of an alloy is generally higher than that of its constituent metals.

(6) Alloys are used in electrical heating devices, like electric iron, toasters etc because they do not oxidise (burn) readily at high temperatures.

(7) Tungsten is used almost exclusively for filaments of electric bulbs, because it does not melt at higher temperature, have high resistivity and can be easily drawn in to thin wires. 

(8) Copper and aluminium are generally used for electrical transmission lines because of their low resistivity they behave as a good conductor of electricity.

NOW CHECK YOUR PROGRESS!!!

1. A wire of given material having length l and area of cross-section A has a resistance of 4 Ω. What would be the resistance of another wire of the same material having length l/2 and area of cross-section 2A?

2. A wire of resistance 20 Ω is stretched to double its length. What will be its new (i) resistivity (ii) resistance?

3. On what factors does the resistance of a conductor depend?

4. Define the SI unit of resistance.

5. Why is the tungsten used almost exclusively for filament of electric lamps?

6. Why are the conductors of electric heating devices, such as bread-toasters and electric irons, made of an alloy rather than a pure metal?

7. How does the resistance of a wire vary with its area of cross-section?

8. Why copper and aluminium wires are usually employed for electricity transmission?

9. Give two examples of materials which are (i) good conductor (ii) resistor (iii) insulator (iv) poor conductor.

10. Will current flow more easily through a thick wire or a thin wire of the same material, when connected to the same source? Why?

11. Why do electricians wear rubber hand-gloves while working with electricity?

12. Why are coils of electric toasters and electric irons made of an alloy rather than a pure metal?

13. Name the device used to change resistance in a circuit to regulate current without changing the voltage source.

14. What is nichrome? State its one property and one use.

15. On what factor does the resistivity of a material depends?

Electric Potential

ELECTRIC POTENTIAL 

The movement of electrons in a metal wire takes place only if there is some sort of difference in electric pressure – called the potential difference – along the conductor. This difference of potential may be produced by a battery. The chemical action within a cell generates the potential difference across the terminals of the cell, even when no current is drawn from it. When the cell is connected to a conductor, the potential difference sets the charges in motion in the conductor and produces an electric current. 

Definition of Electric Potential: 

We define the electric potential difference between two points in an electric circuit as the work done to move a unit charge from one point to the other. 

Formula:

\begin{equation} \textit{ Potential difference between two points}  = \frac{work\ done}{charge} \end{equation}

Or in terms of symbols, we can write

\[V=\frac{W}{Q}\]

SI unit: 

The SI unit of electric potential difference is volt (V), named after Alessandro Volta. 

Definition of one volt: 

One volt is the potential difference between two points when 1 joule of work is done to move a charge of 1 coulomb from one point to the other. 
Therefore, 

\[1 volt=\frac{1 joule}{1 coulomb}\]

Device used to measure potential difference: 

The potential difference is measured by means of an instrument called the voltmeter.

Symbol in an electric circuit:

Connection: 

The voltmeter is always connected in parallel across the points between which the potential difference is to be measured. 

NOW CHECK YOUR PROGRESS!!! 

1. Name a device that helps to maintain potential difference across a conductor. 

2. What is meant by saying that potential difference between two points is 1V? 

3. Write the relation which states the relation between potential difference and work done. 

4. How much energy is given to each coulomb of charge passing through a 6V battery?

5. How much work is done in moving a charge of 2 C across two points having a potential difference of 12 V? 

6. Name the device that measures the potential difference across two points in an electric circuit. How it is connected in an electric circuit?