Resistance

We know electric current is the flow of electrons. When the electrons move within the bulk of a conductor, they collide with the atoms and molecules of the conductor. Due to this their motion is resisted and electric current is obstructed. This property of a conductor is called resistance.

At particular temperature,

Resistance, R

=V ÷ I

= Potential difference of two ends ÷ Electric current

The SI unit for resistance is ohm. It is expressed by the capital letter omega (Ω).

1 Ω is the resistance of a conductor through which a current of 1A flows when a potential difference of 1 V is applied across it.

Resistors

A resistor is a conductor used in a circuit that has a known value of resistance. The main objective of using resistors is to control the quantity of the current flowing in a circuit. There are two types of resistors that are used in a circuit. These are:

1. Fixed resistors

2. Variable resistors

Fixed resistors

The fixed resistors are those who have fixed values of resistance. The fixed resistors that are generally used in laboratory are shown in figure- Fixed resistors.

Fixed resistors

Variable resistors

The variable resistors are those whose value of the resistance can be changed according to the necessity. These are called rheostat too. A rheostat is included in a circuit to vary the current flowing through it. Figure- Variable resistors shows a rheostat commonly used in the laboratories.

Variable resistors

Dependence of resistance

We know, when temperature and other physical conditions (e.g. - length, cross section, area) remain the same, the resistance of a conductor is a constant. The resistance of a conductor depends on four factors.

1. Length of the conductor.

2. Cross sectional area of the conductor.

3. Materials of the conductor and.

4. Temperature of the conductor.

We know, if the temperature remains constant, the resistance of a conductor depends only on its length, area of cross section and the material of the conductor. This dependence of resistance is expressed by two laws.

Figure- 1 shows two conducting wires P and Q with the same cross sectional area and made of the same material. The length of wire P is longer than that of Q. Its resistance is greater as it is longer.

Figure- 1

Law of length

The resistance of a conductor is directly proportional to its length when the cross sectional area, material and temperature of the conductor remain the same. If the length of the conductor is L, area of cross section is A, and its resistance is R, then according to this law,

R ∝ L, when temperature, material and A is constant.

Figure- 2

Figure- 2 shows two conducting wires S and T with the same length and made of the same material. The area of cross section of wire S is bigger than the area of cross section of wire T. Larger the area of cross section of a wire, the lower its resistance.

Law of cross section

The resistance of a conductor is inversely proportional to its cross sectional area when the length, material and temperature of the conductor remain the same.

That is, R ∝ I ÷ A, when temperature, material and L is constant

Resistivity and Conductivity

At constant temperature, the resistance of a conductor of particular material varies proportionately with the length and inversely with the area of cross section. Therefore, we get from the laws of resistance,

R ∝ L ÷ A, when temperature and material remain the same

Or, R =ρ L ÷ A - - - - (i)

Here ρ is a constant, the value of which depends on the material of the conductor and its temperature. This constant is called the resistivity or specific resistance of the material at that temperature.

In equation (i), if L=1 unit, A=1 unit, then, ρ = R.

Therefore, at a particular temperature, the resistance of a conductor of unit length and unit cross sectional area is called the specific resistance of that material at that temperature.

At a certain temperature, the resistance of a conductor depends on its physical conditions (e.g. length, cross section etc.). But the resistivity of a conductor depends only on its material.

Unit of specific resistance

Rewriting equation (i) we can write,

ρ = R A ÷ L

Substituting the units of the quantities on the right side of the equation, the unit of ρ is

Ωm² ÷ m = Ωm

Significance: The resistivity of silver at 20 °C is 1.6×10-8 Ω m. Therefore, the resistance of a silver wire of length 1m and cross sectional area of 1m2 is 1.6×10-8 Ω. Table shows the values of the resistivity of some common materials.

Table of Resistivity’s of different materials

Material	Resistivity (Ω m)
Silver	1.6×10-8
Copper	1.7×10-8
Tungsten	5.5×10-8
Nichrome	100×10-8

From the table above we see that the materials with lower resistivity are good conductors of electricity. For example- copper is much better conductor of electricity than nichrome. Due to this, copper is widely used as connecting wires in electrical circuits.

Besides, materials with higher resistivity also have multiple uses. One example is the

nichrome wire. The resistivity and melting point of nichrome is much higher than that of copper. Due to the high resistivity of nichrome, a lot of thermal energy is produced when a current flows through it. This property of nichrome causes water to boil very quickly in electric kettle. The filament of electric bulbs that are used in our houses is made of tungsten. Tungsten can convert electrical energy to light and thermal energy owing to its high resistivity and melting point.

Conductivity

The reciprocal quantity of resistance is called conductance. Like that, the reciprocal quantity of specific resistance is called conductivity. Conductivity is expressed by the letter σ. The value of σ depends on the type of material of the conductor and its temperature.

Say, the specific resistance of the material of a conductor = ρ

Therefore, the conductivity of its material is σ = 1 ÷ ρ

As unit of ρ is Ω m, Therefore, the unit of σ is (Ω m)^-1.

Making series and parallel circuits and their uses

Series circuit

Series circuit

The circuit in which the electric components are connected one after another in a single loop is called a series circuit. By arranging a cell E, two bulbs B1 and B2 one after another a series circuit is formed in figure- Series circuit. As there is a single path in the circuit, the same current will flow throughout the whole circuit. Now if the ammeter is connected at the points A, B or C, the value of the electric current will found to be the same.

The little bulbs that are used for decoration purpose in wedding ceremony or in different programs are connected in series. We increase the voltage by connecting more than one battery in series in a torch light. The ammeter is connected in series to measure the electric current in a circuit.

Parallel circuit

Parallel circuit

The circuit in which the electric components are arranged in such a way that one terminal of all the components are joined at a common point and the other terminal are joined at another common point then this circuit is called a parallel circuit. In figure- Parallel circuit one end of the bulbs B1 and B2 are connected at the point and the other end at the point b and so formed a parallel circuit. In a parallel circuit there are alternative paths for the current to flow.

Say, the total current in the circuit is I which splits into two parts I1 and I2 at the junction a. Let I1 and I2 are the currents flowing through the bulbs B1 and B2 respectively. At the junction b the currents I1 and I2 recombine to form the current I again. If the current at the points P, Q and R is measured by an ammeter, then it will be found that

I = I1 + I2

Here, total current of the circuit = I

I.e. in a parallel circuit, the sum of the individual currents flowing through each of the parallel branches is equal to the total current.

The electrical appliances such as- light, fan etc. which we use in houses or offices is connected in parallel to the AC mains. Each of the appliances gets the same voltage supply due to parallel connection. But they get different amounts of current.

Equivalent resistance and its uses in circuit

Sometimes several resistances are connected together for different purposes. Connection of more than one resistance together is called combination of resistances.

Equivalent resistance: If a single resistance is used instead of combination of resistances and if the current and potential difference is not changed in the circuit, then that resistance is called the equivalent resistance of the combination.

Combination of resistances is of two types, e.g. - series combination and parallel combination.

Series combination of resistances

Figure shows resistors R1, R2 and R3 are connected in series. The resistances are connected one after another successively. In this case, the same current I is flowing through each of the resistors. Now we shall calculate the equivalent resistance of these three resistances those are connected in series.

Series combination of resistances

From Ohm’s law we get,

The potential difference across resistance R₁, V₁ = IR₁

The potential difference across resistance R₂, V₂ = IR₂

The potential difference across resistance R₃, V₃ = IR₃

If V is the potential difference between the two terminals of all the resistors, i.e. the potential difference across the combination,

∴ V =V₁ +V₂ +V₃

= IR₁ + IR₂ + IR₃

= I (R₁ + R₂ + R₃)

Now if three resistances R₁, R₂ and R₃ are replaced by a single resistance R_s , so that same current I flows through the circuit and the potential difference V across them remains unchanged, then R_s is the equivalent resistance of the combination.

In case of equivalent resistance, V = IR_s

Comparing equations we get,

 IR_s = I (R₁ + R₂ + R₃ )

R_s = R₁ + R₂ + R₃

If instead of three resistances, n numbers of resistances are connected in series then equivalent resistance R_s  will be

R_s = R₁ + R₂ + R₃ + . . . .  + R_n

Therefore, the equivalent resistance of resistors connected is series is equal to the sum of the different resistances included in the combination. The value of the equivalent resistance in series combination is greater than that of individual resistances.

Parallel combination of resistances

When several resistances are connected in such a way that one terminal of all the resistances are joined at a common point A and the other terminals are joined at another common point B and potential difference across each of the resistors remains the same, then this combination of resistances are called parallel combination of resistances.

Parallel combination of resistances

Three resistors R₁, R₂ and R₃ are connected in a parallel combination. In this case, same potential difference V is maintained across the two terminals of the three resistors. Different amount of current is flowing through each of the resistors owing to their different values. The main current I of the circuit splits into three parts at the junction a and later recombine at the point b. Let I₁, I₂ and I₃ are the currents flowing through the resistances R1, R2 and R3 respectively. Therefore, sum of the currents I₁, I₂ and I₃ of parallel paths is equal to the current I at the junction a. Therefore,

I = I₁ + I₂ + I₃ - - - - (ii)

Here, the potential difference between the two terminals being V, applying Ohm’s law we get,

I₁ = V ÷ R₁ ,

I₂ = V ÷ R₂ ,

And I₃ = V ÷ R₃ ,

Substituting the values of I₁, I₂ and I₃ in equation (ii) we get,

I = (V ÷ R₁) + (V ÷ R₂) + (V ÷ R₃)

= V {(1 ÷ R₁) + (1 ÷ R₁) + (1 ÷ R₁)} - - - - (iii)

Now if three resistances R₁, R₂ and R₃ are replaced by a single resistance R_P, so that same current I flow through the circuit and the potential difference V across them remains unchanged, then R_P is the equivalent resistance of the combination.

∴ I = V ÷ R_p - - - - (iv)

Comparing equations (iii) I (iv) we get,

V ÷ R_p = V {(1 ÷ R₁) + (1 ÷ R₁) + (1 ÷ R₁)}

1 ÷ R_p = {(1 ÷ R₁) + (1 ÷ R₁) + (1 ÷ R₁)}

If instead of three resistances, n numbers of resistances are connected in parallel then the equivalent resistance RP can be expressed as

1 ÷ R_p = (1 ÷ R₁) + (1 ÷ R₁) + (1 ÷ R₁) + . . . . + (1÷R_n)

That is, resistances connected in parallel combination, the sum of the inverse of the individual resistances is equal to the inverse of the equivalent resistance.

End

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