RESISTIVITY AND ITS DEPENDENCE UPON TEMPERATURE

It has been experimentally seen that the resistance R of a wire is directly proportional to its length L and inversely proportional to its cross sectional area A. Expressing mathematically we have

R ∝ L 

And 

R ∝ 1/A 

Or 

R ∝ L/A 

R = ρ L/A ....... (1)

where ρ is a constant of proportionality known as resistivity of specific resistance of the material of the wire. It may be noted that resistance is the characteristic of a particular wire whereas the resistivity is the property of the material of which the wire is made. From Eq. 1 we have

ρ = RA/L

The above equation gives the definition of resistivity as the resistance of a metre cube of a material. The SI unit of resistivity is (Ωm).

Conductance is another quantity used to describe the electrical properties of materials. In fact conductance is the reciprocal of resistance i.e.,

Conductance = 1 / resistance (R)

The SI unit of conductance is mho or siemen.

Likewise conductivity, σ is the reciprocal of resistivity i.e., 

σ = 1/ρ

The SI unit of conductivity is ohm^-1m^-1 or mho m^-1. Resistivity of various materials are given in Table.

It may be noted from Table that silver and copper are two best conductors. That is the reason that most electric wires are made of copper.

The resistivity of a substance depends upon the temperature also. It can be explained by recalling that the resistance offered by a conductor to the flow of electric current is due to collisions, which the free electrons encounter with atoms of the lattice. As the temperature of the conductor rises, the amplitude of vibration of the atoms in the lattice increases and hence, the probability of their collision with free electrons also increases. One may say that the atoms then offer a bigger target, that is, the collision cross-section of the atoms increases with temperature. This makes the collisions between free electrons and the atoms in the lattice more frequent and hence, the resistance of the conductor increases.

Experimentally the change in resistance of a metallic conductor with temperature is found to be nearly linear over a considerable range of temperature above and below 0 °C. Over such a range the fractional change in resistance per kelvin is known as the temperature coefficient of resistance i.e..

 α = (R_t - R₀)/R₀t

where R₀, and R_t, are resistances at temperature 0 °C and t°C. As resistivity ρ depends upon the temperature, Eq. gives 

R=ρ₀ L/A and R=ρ_t L/A

Substituting the values of R₀, and R_t, in Eq. we get

α = (ρ_t - ρ₀)/ρ₀t

where ρ₀ is the resistivity of a conductor at 0 °C and ρ_t is the resistivity at t °C. 

There are some substances like germanium, silicon etc; whose resistance decreases with increase in temperature. i.e;  these substances have negative temperature coefficients.

OHM'S LAW

We have seen that when a battery is connected across a conductor, an electric current begins. 

How much current flows through the conductor when a certain potential difference is set up across its ends? 

The answer to this question was given by a German Physicist George Simon Ohm. 

He showed by elaborate experiments that the current through a metallic conductor is directly proportional to the potential difference across its ends. This fact is known as Ohms' law which states that

"The current flowing through a conductor is directly proportional to the potential difference across its ends provided the physical state such as temperature etc of the conductor remains constant."

Symbolically Ohm's law can be written as

Ι ∝ ν

It implies that

V=RI

where R, the constant of proportionality is called the resistance of the conductor. 

The value of the resistance depends upon the nature, dimensions and the physical state of the conductor.  

Define resistance.

The resistance is a measure of the opposition to the motion of electrons due to their continuous bumping with the atoms of the lattice. 

Unit of resistance.

The unit of resistance is ohm. 

One ohm.

A conductor has a resistance of 1 ohm if a current of 1 ampere flows through it when a potential difference of 1 volt is applied across its ends. 

The symbol of ohm is Ω. 

If I is measured in amperes, V in volts, then R is measured in ohms i.e.

R(ohms)=V (volts)/I (amperes)

Ohmic Conductors

A sample of a conductor is said to obey Ohm's law if its resistance R remains constant that is, the graph of its V versus I is exactly a straight line. A conductor which strictly obeys Ohm's law is called ohmic. 

Non Ohmic Conductors

There are devices, which do not obey Ohms' law i.e., they are non ohmic. The examples of non ohmic devices are filament bulbs and semiconductor diodes.

Let us apply a certain potential difference across the terminals of a filament lamp and measure the resulting current passing through it. If we repeat the measurement for different values of potential difference and draw a graph of voltage V versus current 7, it will be seen that the graph is not a straight line. It means that a filament is a non ohmic device. This deviation of I-V graph from straight line is due to the increase in the resistance of the filament with temperature.

As the current passing through the filament is increased from zero, the graph is a straight line in the initial stage because the change in the resistance of the filament with temperature due to small current is not appreciable. As the current is further increased, the resistance of the filament continues to increase due to rise in its temperature.

Another example of non ohmic device is a semiconductor diode. The current - voltage plot of such a diode is shown in Figure. The graph is not a straight line so semi conductor is also a non ohmic device.

Review of Series and Parallel Combinations of Resistors

In an electrical circuit, usually, a number of resistors are connected together. There are two arrangements in which resistors can be connected with each other., one is known as series arrangement and other one as parallel arrangement.

Series Combination of resistance 

If the resistors are connected end to end such that the same current passes through all of them, they are said to be connected in series as shown in Figure. There equivalent resistance R, is given by

R_eq=R₁+R₂+R₃+------

Parallel combination of resistance 

In parallel arrangement a number of resistors are. connected side by side with their ends joined together at two common points as shown. in Figure. The equivalent resistance R. of this arrangement is given by 

1/R_eq = 1/R₁ + 1/R₂ + 1/R₃+..........

EFFECTS OF CURRENT

The presence of electric current can be detected by the various effects which it produces. The obvious effects of the current are:

(i) Heating effect

(ii) Magnetic effect

(iii) Chemical effect 

Heating Effect

Current flows through a metallic wire due to motion of free electrons. During the course of their motion, they collide frequently with the atoms of the metal. At each collision, they lose some of their kinetic energy and give it to atoms with which they collide. Thus as the current flows through the wire, it increases the kinetic energy of the vibrations of the metal atoms. i.e., it generates heat in the wire. It is found that the heat H produced by a current / in the wire of resistance R during a time interval tis given by

H=I²Rt

The heating effect of current is utilized in electric heaters, kettles, toasters and electric irons etc. 

Magnetic Effect

The passage of current is always accompanied by a magnetic field in the surrounding space. The strength of the field depends upon the value of current and the distance from the current element. The pattern of the field produced by a current carrying straight wire, a coil and a solenoid is shown in Figure. Magnetic effect is utilized in the detection and measurement of current. All the machines involving electric motors also use the magnetic effect of current.

Chemical Effect

Certain liquids such as dilute sulphuric acid or copper sulphate solution conduct electricity due to some chemical reactions that take place within them. The study of this process is known as electrolysis. The chemical changes produced during the electrolysis of a liquid are due to chemical effects of the current. It depends upon the nature of the liquid and the quantity of electricity passed through the liquid. 

The liquid which conducts current is known as electrolyte. The material in the form of wire or rod or plate which leads the current into or out of the electrolyte is known as electrode. The electrode connected with the positive terminal of the current source is called anode and that connected with negative terminal is known as cathode. The vessel containing the two electrodes and the liquid is known as voltameter. As an example we will consider the electrolysis of copper sulphate solution. The voltameter contains dilute solution of copper sulphate. The anode and cathode are both copper plates. 

When copper sulphate is dissolved in water, it dissociates into Cu⁺⁺ and SO₄^--, ions. On passing current through the voltameter. Cu moves towards the cathode and the following reaction takes place.

Cu⁺⁺+2e^-- -------> Cu 

The copper atoms9 thus formed are deposited at cathode plate. While copper is being deposited at the cathode, the SO, ions move towards the anode. Copper atoms from the anode go into the solution as copper ions which combine with sulphate ions to form copper sulphate. 

Cu+SO -----> CuSO

As the electrolysis proceeds, copper is continuously deposited on the cathode-while an equal amount of copper from the anode is dissolved into the solution and the density of copper sulphate solution remains unaltered.

This example also illustrates the basic principle of electroplating a process of coating a thin layer of some expensive metal (gold, silver etc.) on an article of some cheap metal.

SOURCE OF CURRENT

When two conductors at different potentials are joined by a metallic wire, current will flow through the wire. The current continues to flow from higher potential to the lower potential until both are at the same potential. After this the current ceases to flow. Thus the current through the wire decreases from a maximum value to zero. In order to have a constant current the potential difference across the conductors or the ends of the wire should be maintained constant. This is achieved by connecting the ends of the wire to the terminals of a device called a source of current.

Every source of current converts some non electrical energy such as chemical, mechanical, heat or solar energy into electrical energy. There are many types of sources of current. A few examples are mentioned below:

(1) Cells (primary as well as secondary) which convert chemical energy into electrical energy.

(ii) Electric generators which convert mechanical energy into electrical energy.

(iii) Thermo-couples which convert heat energy into electrical energy.

(iv) Solar cells which convert sunlight directly into electrical energy. 

ELECTRIC CURRENT

An electric current is caused by the motion of electric charge. 

If a net charge ∆Q passes through any cross section of a conductor in time ∆t, we say that an electric current I has been established through the conductor where 

   I = ∆Q/∆t

The SI unit of current is ampere and it is a current due to flow of charge at the rate of one coulomb per second.

Motion of electric charge which causes an electric current is due to the flow of charge carriers. 

• In case of metallic conductors, the charge carriers are electrons. 
• The charge carriers in electrolyte are positive and negative ions.
• In gases, the charge carriers are electrons and ions.

Current Direction

History: Early scientists regarded an electric current as a flow of positive charge from positive to negative terminal of the battery through an external circuit. 

Later on, it was found that a current in metallic conductors is actually due to the flow of negative charge carriers called electrons moving in the opposite direction i.e., from negative to positive terminal of the battery, but it is a convention to take the direction of current as the direction in which positive charges flow. 

This current is referred as conventional current. The reason is that it has been found experimentally that positive charge moving in one direction is equivalent in all external effects to a negative charge moving in the opposite direction. 

As the current is measured by its external effects so a current due to motion of negative charges, after reversing its direction of flow can be substituted by an equivalent current due to flow of positive charges.

Conventional Current 

"The conventional current in a circuit is defined as that equivalent current which passes from a point at higher potential to a point at a lower potential as if it represented a movement of positive charges."

While analyzing the electric circuit, we use the direction of the current according to the above mentioned convention. 

If we wish to refer to the motion of electrons, we use the term electronic current. 

Current Through a Metallic Conductor

In a metal, the valence electrons are not attached to individual atoms but are free to move about within the body. These electrons are known as free electrons. The free electrons are in random motion just like the molecules of a gas in a container and they act as charge carriers in metals. The speed of randomly moving electrons depends upon temperature.

If we consider any section of metallic wire, the rate at which the free electrons pass through it from right to left is the same as the rate at which they pass from left to right. As a result the current through the wire is zero. 

If the ends of the wire are connected to a battery, an electric field E will be set up at every point within the wire. The free electrons will now experience a force in the direction opposite to E. As a result of this force the free electrons acquire a motion in the direction of -E. It may be noted that the force experienced by the free electrons does not produce a net acceleration because the electrons keep on colliding with the atoms of the conductor. The overall effect of these collisions is to transfer the energy of accelerating electrons to the lattice with the result that the electrons acquire an average velocity, called the drift velocity in the direction of -E. 

The drift velocity is of the order of 10^-3 ms^-1, whereas the velocity of free electrons at room temperature due to their thermal motion is several hundred kilometres per second.

Thus, when an electric field is established in a conductor, the free electrons modify their random motion in such a way that they drift slowly in a direction opposite to the field. In other words the electrons, in addition to their violent thermal motion, acquire a constant drift velocity due to which a net directed motion of charges takes place along the wire and a current begins to flow through it. 

Steady Current

A steady current is established in a wire when a constant potential difference is maintained across it which generates the requisite electric field E along the wire.

Electric Charge

Electric Charge is the Intrinsic property of fundamental particles (like electrons, w bosons and quarks etc).

OR

 Electric charge is the intrinsic physical property of matter that causes it to experience a force when kept in an electric or magnetic field. 

An electric charge is associated with an electric field, and the moving electric charge generates a magnetic field. A combination of electric and magnetic fields is known as the electromagnetic field. 

Electrostatics

• The study of charges at rest is called electrostatics.
• The fundamental law of electrostatics is called Coulomb's law.

Current and Electricity 

• The study of charges with constant speed is called current and electricity.
• The fundamental law of current and electricity is Ohm's law. 

Electricity and Magnetism 

• The study of charges with variable speed is called electricity and magnetism. 
• The fundamental laws of electricity and magnetism are Maxwell Equations.

ELECTRIC FIELD

 Transmission of Electric Field 

To describe the mechanism by which electric force is transmitted, Michael Faraday (1791-1867) introduced the concept of an electric field. According to his theory, it is the intrinsic property of nature that an electric field exists in the space around an electric charge. 

Force Field

The electric field is considered to be a force field that exerts a force on other charges placed in that field. For example, a charge q produces an electric field in the space surrounding it. This field exists whether the other charges are present in space or not. 

Presence of Electric Field 

The presence of field cannot be tested until another charge q0, is brought into the field. Thus the field of charge q interacts with q0, to produce an electrical force. The interaction between q and q0, is accomplished in two steps: 

(a) the charge q produces a field 

(b) the field interacts with charge q0, to produce a force F on q. 

These two steps are illustrated in Figure.

In this figure the density of dots is proportional to the strength of the field at the various points. We may define electric field strength or electric field intensity E at any point in the field as where F is the force experienced by a positive test charge q. placed at the point. The test charge q0, has to be very small so that it may not distort the field which it has to measure.

E = F/q0 ......... (1)

Since electric field intensity is force per unit charge, it is measured in newton per coulomb in SI units. It is a vector quantity and its direction is the same as that of the force F.

Place a positive test charge q0, at this point. The Coulomb's force that this charge will experience due to q is

where r is a unit vector directed from the point charge q to the test point where q0, has been placed, i.e., the point where the electric intensity is to be evaluated. By Eq.

Baryons

 

Baryons are strongly interacting particles having half-integral spins (1/2, 3/2, 5/2 . . . . .). A partial listing of some baryons is given in Table. The familiar members of the baryon family are the proton and neutron. Baryons have distinct antiparticles for example, the antiproton (p) and antineutron (n).

Some selected mesons

Particle	Antiparticle	Charge	Spin (h/2π)	Strangeness	Rest Energy (MeV)	Mean Life (s)	Typical Decay Products
p		+1	½	0	938	ꝏ
n		0	½	0	940	887	p + e- + e
A0		0	½	-1	1116	2.6 x10-10	p + π-
Σ+		+1	½	-1	1189	0.8x10-10	p + π0
Σ0		0	½	-1	1193	7.4x10-20	A0 + ϒ
Σ -		-1	½	-1	1197	1.5x10-10	n + π-
Ξ0		0	½	-2	1315	2.9x10-10	A0 + π0
Ξ-		-1	½	-2	1321	1.6x10-10	A0 + π-
∆*		+2,+1 0,-1	3/2	0	1232	5.9x10-24	p + π
Σ*		+1,0,-1	3/2	-1	1385	1.8x10-23	A0 + π
Ξ*	---	-1, 0	3/2	-2	1530	7.3x10-23	Ξ + π
Ω-	---	-1	3/2	-3	1672	8.2x10-11	A0 +K-
Ae+	---	+1	½	0	2285	2.1x10-13	p +K- + π+
Ab0	---	0	½	0	5624	1.2x10-12	p +D0 + π-

We can produce heavier baryons in reactions between nucleons, such as 

p + p——> p +A^o  + K⁺,

which produces the A^o baryon and the K⁺ meson. The A^o decays according to

A^o ——> p + π^-          (mean life = 2.6 X 107" s).

Although there are no neutrinos produced in the decay, the

mean life indicates that the decay is governed by the weak

interaction.