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Current Electricity

 

Electric Current :

    The current ‘i‘ is same for all cross-sections of a conductor, even though the cross-sectional area may be different at different points. This constancy of electric current follows because charge must be conserved it does not pile up steadily or drone away steadily from any point in the conductor under the assumed steady state conditions.

    The existence of electric field inside a conductor does not contradict E = 0 inside the conductor condition of electrostatics. In electrostatics we dealt with a static in which all net motion of charge had stopped and assumed that conductor was insulated and no potential difference applied across it. In this there is no such condition. The direction of current does not signify at all that current is a vector quantity because the current does not obey the rules of vector addition. Charge carriers can only move along the wire. They do not have any direction of their own.

    We know that potential is the property which determines the direction of flow of current. If we have two bodies placed certain distance apart at different potential. If there is no connection between them their potential remains constant but once they are connected, charge (positive) flows from body at higher potential unless their potential becomes equal.

    But this is an instantaneous process and completes in a very short interval of time. To maintain current flow in between two bodies continuously we have to connect them across with a source of emf which helps in maintaining potential difference between them and current is defined as the rate of flow of charge across a cross-section of the conductor.


    This relation holds good for uniform charge flow. For non-uniform charge flowing across any cross section of the conductor the relation will be,


Units of electric current :

(i)    In SI, the current flow is measured in Ampere and current is said to be one ampere if one coulomb of charge flows across any cross section in one second.


(ii)    In cgs (e.s.u.) system, the current flowing across any cross-section is measured in stat-ampere and


(iii)    In cgs (electromagnetic unit) system, the current flowing through a conductor is said to be measured in ab-ampere.


    

Relation between units of current :


Conventional current :

    By convention, the direction of flow of current is taken to be the direction of flow of positive charge. The electrons always flow in direction opposite to marked in circuit diagrams.

Charge carriers in Substances :

In Solids :

    In solids the charge carriers are the free electrons. As we know that in certain substances outermost orbits are very loosely bound to nucleus due to their large distance from nucleus. Hence these electrons are assumed to be free electrons. When potential difference is applied across two ends of such a conductor electrons flow from lower to higher potential.

In Liquids :

    Liquids are classified into two categories depending upon their electrical properties. Electrolytes are the liquids which conduct electricity and non-electrolytes are liquids which do not conduct electricity or we can say that liquid which can break into their ionic form are electrolyte or conductors and liquids which do not break into ionic form are insulators. The charge carrier in electrolytes are the positive and negative ions.

In Gases :

    Gases are normally insulators and do not conduct due to absence of any kind of free charges in them. But when the gas gets ionised due to passage of X rays etc., the positive and negative ions carry the charges. In gases if X rays strike the gas atom it knocks out an electron from it. The electrons thus acts as negative ion and which loses electrons acts as positive ion.

 

Type of currents :

    The electric current can be classified into the following categories :

(i)    Steady current : The current whose magnitude remains constant or does not change with time is called steady current. As shown in figure curve [A] represents steady current

(ii)    Varying current : The current whose magnitude does not change with time is called varying current. Alternating current is an example of varying current. Curve [B] and [C] represents varying current.

 

Drift Speed :

    A conductor contains large number of loosely bound electrons which we call free electrons or conduction electrons. The remaining material is collection of heavy positive ions called lattice. These ions keeps on vibrating about their mean positions. The average amplitude of vibration depends upon temperature. Occasionally, a free electron collides interacts in some other fashion with the lattice. The speed and direction of electron changes randomly at each such event. As a result electrons moves in a zig-zag path. As there is a large number of free electrons moving in random directions, the number of electrons crossing unit are DS from side nearly equals the number crossing from other side in any given interval. The electric current through the area is therefore zero.

    When there is an electric field inside the conductor, a force acts on each electron in a direction opposite to the field. The electrons get biased in their random motion in favour of the force. As a result the electrons drift slowly in this direction. At each collision electrons starts afresh in random direction with random speed but gains an additional velocity v¢ due to electric field. This velocity v¢ increases with time and suddenly becomes zero as the electron makes collision with the lattice and starts afresh with random velocity. As a result time t between successive collisions is small, the electrons slowly and steadily drifts opposite to applied field. If the electron drifts a distance ‘l‘ in long time t, we define drift speed as


    The average time between two successive collisions is called the relaxation time. If t is the relaxation time, average distance drifted during this period is


The drift speed is thus


    The constant k depends upon the material of the conductor and its temperature. Consider a cylinderical conductor with area of cross-section ‘a‘ and charge density (i.e. number of electrons per unit volume) be ‘n‘. In time dt electrons will transverse a length equal to vddt.

The total number of charges in this length of conductor = n A vd dt

Total charge in this length = n e A vd dt = dq


I = n e A vd

and current density


    The direction of the drift velocity of a positive charge is the same as that of the electric field and the direction of velocity of negative charge is opposite to . Thus even in a metallic conductor where moving charges are negative electrons only and move in opposite direction to , the vector current density is in the same direction as .

 

Ohm’s Law :

    It states that if physical conditions remain unchanged, he current flowing through a conductor is always directly proportional to the potential difference applied across conductor.

V a I

V = IR

Where R is constant of proportionality called resistance of the conductor. Resistance is a quantity, which depends upon the nature of material and its dimensions. Although resistance id independent of V and I. For conductors, which obey Ohm’s law, V- I graph is a straight line as shown in the figure.

Units of resistance: mathematically resistance is the ratio of potential difference applied across conductor to the current flowing through it.. thus, its SI units are volt/ampere or commonly called ohm[W].

1 ohm =

Thus, resistance of the conductor is said to be I ohm, when unit potential difference [i.e volt] applied across conductor results in unit current [i.e. 1 ampere] through it.

Cause of resistance: The resistance of conductor physically implies opposition to the flow of current. When potential difference is applied across the conductor the electros gets accelerated. But as they accelerate, they collide with other atoms and ions and their motion is opposed. This opposition is called resistance of the conductor.

Non- Ohmic Conductor: the conductor which don’t obey ohm’s law or which V-I curve is not a straight line are called non-ohmic conductors. Even for non-ohmic conductors relation = IR can be used but for such conductors, R will not be constant s in the case of ohmic conductors. For e.g. thermistors, thyristor, diode etc.

Failure’s Of Ohmic Law: The few cases where ohm’s law is not obeyed is

[a] Potential difference may vary non-linearly with current: When current flows through a conductor, the temperature of the conductor begins to increase due to heating effect of the current. As the temperature rises the resistance of the conductor increases. Increase in resistance with temperature results n V-I deviating from straight line.

[b] Variation of current with potential difference applied depends upon the direction of electric field inside the conductor: This happens generally in the case of semiconductor diodes which allow one way flow of current. When positive terminal of battery is connected with p type and negative with n type resistance is small and when connections are reversed resistance is large

Color coding of Resistance :

    Commercial resistances available in market have their magnitude written in the form of color codes. The two color code systems used are:

(i)    We associate a color with each digit, 0, 1, . . . . . ., 9

 

Black = 0, Brown = 1, Red = 2, Orange = 3, Yellow = 4, Green = 5, Blue = 6, Violet = 7, Gray = 8, White = 9

 

    The three coloured bands on one side indicate its resistance. The first two bands [A and B] from one end indicate the corresponding digits while third band’s color indicate powers of ten with which number must be multiplied to get the resistance value in ohm.

    In addition to three bands, fourth band gives us tolerance with silver band implying tolerance of ± 5% and gold band with tolerance of ± 10%. No fourth band indicates resistance of ± 20%. For example, if four bands are of yellow, red, blue and gold then resistance will be (42 ´ 106) ± 5%

(ii)    In second system, the body of the resistance is of one color, the end are given another color while a dot is marked over the body. A ring on one side determines tolerance. The colour code remains same.

 

Factors affecting the Resistance of Wire (Resistivity) :

    The resistance of material depends upon nature of conductor, its shape and size. Simple considerations show that: Resistance of conductor is directly proportional to length and inversely proportional to the area of cross-section of conductor.


    where r is the constant of proportionality called the resistivity of the material. Its value is constant for a given material, independent of shape and size of the conductor. As r =, therefore if length and area are changed resistance of the conductor changes in such a fashion that the resistivity remains constant. The units of r are .

NOTE : As potential difference is applied across length of conductor, electrons move from lower to higher potential. In the process they collide with each other and their motion is retarded. Now, if the length of the conductor is increased it will experience greater number of collisions and hence the resistance increases.

    Similarly, larger area of cross-section means large number of electrons will cross the cross-section of conductor in one second, thereby giving a larger current. Larger the value of current smaller will be the resistance (according to Ohm’s law)

 

Conductivity:

    Conductivity is defined as the reciprocal of resistivity or specific resistance. Therefore large resistivity implies smaller conductivity and vice-versa. It is generally denoted by s.


It is measured in ohm—1 m—1 or mho/m

 

Conductance :

    The reciprocal of resistance of a conductor is called conductance. It is denoted by C. The conductance


    It is measured in ohm—1 or mho. It is also measured in Siemens.

 

Origin of Electric Resistance :

    When electric field is applied across the conductor, the electrons are accelerated but as they accelerate they collide with positive ions of crystal lattice which retards their motion. Thus, the electrons after some time acquires a constant velocity.

    For a perfect crystal lattice with all positive ions fixed, regularly at specified positions it can be proved quantum mechanically that conduction electron moves freely through the lattice under action of external field. However, no metal is composed of perfect crystal lattice. In some instances the imperfection is due to impurities that replace some of the metal ions. In addition to ions are always vibrating as a result of thermal energy. Since vibrating as a result of thermal energy. Since ions do not vibrate in phase, the distances between ions fluctuate. This fluctuation is equivalent to imperfections in crystal lattice. The electron thus suffers numerous scattering as a result of these imperfections and sometimes even moves backwards. Therefore rather than picking up energy continuously from electric field, the electron transfers some energy to lattice. After a short period of time a steady state is reached where average velocity becomes constant.