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Current Theory part-2

Effect of temperature on Resistance :

    It was experimentally found that resistivity is inversely proportional to the relaxation time, because smaller value of t will mean more frequent collisions and hence larger resistivity. The relaxation is dependent on temperature of the given substance. An increase in temperature will mean that ions and atoms are vibrating more vigorously about their mean position and thus the frequency of collision increases. Hence a rise in temperature for conductors result in increase in resistivity of material. For some substances the increase in resistivity is linear such that we can write,

rt = r0 ( 1 + a t )

where rt and r0 are the resistivities at tºC and 0ºC.

The constant a
is defined by,


is known as linear coefficient of resistivity. It can be defined as change in resistivity per unit original resistivity per degree rise in temperature.

    Even for metals, resistivity increases as higher power of T for low temperatures. For insulators and semiconductors resistivity increases exponentially with decreasing temperature.

r(T) = r0(T) eEg/kT

    where Eg is forbidden gap energy and k is Boltzmann constant. For insulators or semiconductors resistance decreases with increase in temperature because of the number of charge carriers available for conduction. There is an exponential increase in number of charge carriers.

 

Semiconductors:

    In semiconductors the number of charge carriers per unit volume ‘n‘ increases rapidly with increasing temperature. The increase in ‘n’ for outweighs increase in ‘u‘ and the resistivity decreases. At low temperatures, n is very small and resistivity becomes so large that molecule can be considered insulator.

    The modern theory of super conductivity predicts that in effect at temperatures below the critical temperature the electrons moves freely throughout the lattice. The mean free path l then becomes very large and resistivity is very small.

 

Superconductivity:

    We have seen that as the temperature of conductor decreases the resistance also goes on decreasing. It was found by Onnes that mercury possessed zero resistance at temperature of 42K i.e. mercury has become superconductor. Later it was found that all conductors had their resistivity decreased to zero at certain value of temperature called critical temperature of conductor. Thus critical temperature can be defined as the temperature at which resistivity falls to zero and the conductor becomes superconductor.

    A body is said to be superconductor if current flowing through it, keeps on flowing for 1017 years without any decay.

    The critical temperature of pure metals is very low therefore alloys are developed whose critical temperature should be high. In 1987 a ceramic yittrium, barium and copper oxide was found to have TC as 90K.

    The cause of superconductivity is that at low temperature the electrons do not remain independent and move in coherent clouds. Ionic vibrations are unable to deflect cloud of electrons whereas they can defect single electrons. Now, why the clouds start behaving in coherent fashion at low temperatures is yet not known.

 

Uses of Superconductivity:

(i)    It is used for generation of very strong magnetic fields.

(ii)    It is used in cryogenic space technology.

(iii)    It is used in very high speed computers.

    Now, if some day we are able to manufacture superconductors at daily life temperatures (300K – 320K) they can be used to transmit power without energy loss.

 

Relation between Relaxation Time and Resistivity:

    The average time between two collisions is called relaxation time. Consider a conductor placed in electric field, E. The force experienced by electrons present in electric field is, .

The acceleration of electrons            

The drift velocity,                 vd = u + at


Also, if A is the cross-sectional area of the tube then vd is given by

Vd = I/neA



Also E = , Substituting the value we get

As RA/l is the resitivity of the material, thus the Conductivity and resistivity are given by

 


This relation implies that if we increases the temperature of the conductor, the mass number density and charge on electron are constant. But the collisions becomes more freqent. Thus the relaxation time decreases and resistivity increases.

 


 

Resistors in Series and Parallel:

    Most electrical circuits consists not merely of single source and single external resistor but comprise of number of resistors of capacitors etc. interconnected in a complicated manner. Such a circuit is called a network.

    It is always possible to find a single resistor that should replace a combination of resistors in given circuit and leave unaltered the potential difference between the terminals of the combination and the current in rest of the circuit. The resistance of this resistor is called equivalent resistance of the combination.

Series Combination

    If we have number of resistors connected in series combination, the current in each resistor must be same and equal to the line current I, but the potential difference across each resistor is different such that if V1, V2, V3 be the potential difference across R1, R2, R3 then potential difference across series combination is V such that,

V = V1 + V2 + V3

or                        IR = IR1 + IR2 + IR3            (I must be same)

R = R1+R2+R3

For a combination of n resistors

R = R1 + R2 + – – – – – – – + Rn

    The equivalent resistance of any number of resistors connected in series equals the sum of their individual resistances.

PARALLEL combination:

    Resistors are said to be connected in parallel if one end of all the resistors are connected at one point and the second end at the other point. As all the resistors are connected between the same two points therefore potential difference between the terminals of each must be same. If we have number of resistors in parallel then current in each must be,


    Charge is delivered at point a by line current I and removed from a by currents I1, I2, – – – – – , In such that

I = I1 + I2 + – – – – – + In


    Thus, for any number of resistors in parallel, the reciprocal equivalent resistance equals the sum of reciprocals of their individual resistances.

 

Current in Parallel Combination:

    If I be the total current flowing in parallel combination of two resistors R1 and R2, then,

I = I1 + I2

where I1 and I2 are currents in individual resistors R1 and R2 respectively.



Similarly, the current in the second resistance R2 is,


Kirchoff’s Rules:

    These are the rules used for simplifying complicated electrical circuits.

Branch Rule or Junction Rule:

    A junction point in a network is a point where three or more conductors are joined. According to this rule algebraic sum of currents towards a junction point is zero. That is,

S I = 0

Sign Conventions:

    The current metering a junction is said to be positive and current leaving the junction is said to be negative.

    The point rule is an application of conservation of charge. Since no electric charge can accumulate at a branch point the total current entering is always equal to the total current leaving.

 

Loop Rule:

    Loop is any closed path in an electrical circuit. According to the Loop Rule, algebraic sum of all the emf’s is always equal to the sum of product of current and emf i.e.

S E = S IR

Sign Conventions:

a.    Choose any closed loop in the network and designate the direction (clockwise or anticlockwise) to transverse the loop applying Loop Rule.

b.    Go around the loop in the designated direction, adding emf’s and potential differences. An emf is counted as positive when it is transversed from negative to positive and negative otherwise. Similarly, product of current and resistance is positive if the direction of current is same as the direction in which we are transversing the loop, otherwise it is negative.

    These basic rules are used to solve variety of network problems. Usually in problems some of the emf’s currents and resistances are known and others are unknown. The number of equations obtained from kirchoff’s rule must always be equal to the number of unknowns, to permit solution of equations, the following principles should be followed carefully.

    First all quantities known and unknown are labelled carefully, including an assumed sense of direction for each unknown current. Often one does not know in advance the actual direction of current but this does not matter. The solution is carried out using assumed directions the value of quantity will come out to be negative if actual direction is opposite to assumed direction. Hence kirchoff’s laws gives us correct value of unknown current and emf’s.

Ammeters:

    An ammeter is the instrument used for measurement of current in a circuit. Ammeter is essentially a galvanometer which is inserted in the circuit in series so that the whole current in the circuit passes through it. The deflection produced in ammeter is measure of current flowing through it. Since the coil of ammeter has some resistance, so on connecting it in series the resistance of circuit increases and current flowing through it decreases. Therefore, current read by ammeter is less than actual current flowing in the circuit. Thus, resistance of ammeter should be as small as possible, so that connecting it in the circuit does not change the current appreciably. Suppose emf of the cell be E, when connected in circuit having resistance R, current flowing will be


    On connecting ammeter in series in the circuit, the resistance of circuit becomes (R1 + RA) where RA is the resistance of ammeter and current reduces to


    Therefore, deflection of the ammeter shows i¢ although current to be measured was i. For converting galvanometer to ammeter a small resistance called shunt is connected in parallel in the circuit. The combined resistance of the galvanometer and shunt is less than the resistance of the galvanometer and shunt. Therefore, when it is connected in circuit it does not produce an appreciable change in the circuit.

    Another advantage is that when shunt is connected in circuit most of the current tends to pass through shunt and only fraction of total current will pass through galvanometer. Since deflection of coil is proportional to current passing through it, the deflection is sufficiently reduced. Hence now there will be full scale deflection of the coil for much larger current in the circuit.

    Let i be the current flowing in the main current circuit and ig through galvanometer then current through shunt will be,

igG = (i ig) S


Thus, with shunt in circuit galvanometer of range ig is changed into ammeter of range i ampere.

 

Conversion of Galvanometer into Voltmeter:

    Voltmeter is a device used for measurement of potential difference between two points.

    Voltmeter is basically a galvanometer with high resistance connected in series with it and is connected across two points between which p.d. is to be measured. The deflection produced in voltmeter is the measure of p.d. between two points. Since however, it is connected across two points, the potential difference between those points is changed.

    Suppose p.d. is to be measured between two points a and b, before connecting the cell. The current in the circuit is,


    The potential difference between them is


    On connecting voltmeter of resistance RV, the current becomes,


    To eliminate error in calculation R2 = R2¢



or                             RV = ¥

    Thus, ideal voltmeter should have infinite resistance. Suppose a is the resistance of galvanometer and we connected high resistance R in series with it. Suppose on connecting it between points a and b of a circuit, a current ig flows through it. If potential difference between a and b is V.


    If current ig in the coil produces full scale deflection then there will be potential difference V between a and b. Thus on connecting R galvanometer of range ig is changed to a voltmeter of range V.

 

Wheatstone bridge :

    Wheatstone bridge is a special electrical circuit used to determine the value of unknown resistance. It consists of four resistances P, Q, R and S connected across as arms of a parallelogram. In one diagonal a galvanometer is connected in other diagonal a cell. The current i flowing in the circuit gets divided into two parts at A

a)    i flowing through P and (i i1) flowing through R.

b)    Similarly on reaching B, i1 gets divided with ig flowing through galvanometer and (i ig) through Q.

c)    Current flowing in S will be (i i1 + ig).

    Writing loop equation for the loop ABDA,

i1 P + ig G (i i1) R = 0

    Similarly, for the loop BCDB,

(i1
ig) Q (i i1 + ig) S ig G = 0

    Change the value of variable resistance R till B and C are at same potential. Thus, no current will flow through galvanometer or ig = 0. Hence, two equation becomes,

i1 P (i i1) R = 0

and                            i1 Q (i i1) S = 0

or                            

    This is called balanced wheatstone condition. Knowing the value of three resistances, fourth resistance could be easily calculated. The bridge has maximum sensitivity when all the four resistances are of same order.

 

Unsuitability of Wheatstone bridge:

    Four arm wheatstone bridges are best suited for measuring medium resistance. If the resistance are very much high, R1, R2, R3 and R4 should also be high, then current through galvanometer will be small and bridge will not be sensitive.

    If the resistance to be measured is very low, then for bridge to be sensitive all the resistances shall be low. The galvanometer should also be of low resistance which itself is very insensitive. Further the effect of variable contact resistance becomes noticeable and the error due to them may be 0.1 percent or more. This is because, for reason intrinsic to the nature of bridge, the contact resistance being in series with arm resistors are included in measuring network.

 

Meter Bridge:

    Meter bridge is a device used for determination of resistance using wheatstone bridge principle. It consists of 1m long manganin or constanton wire fixed along a scale on a wooden base. The area of cross section of the wire should be uniform throughout. The wire is connected between two copper strips. Another copper strip is fixed between two strips to provide two gaps. A resistance box is connected between one gap and unknown resistance in second gap. A galvanometer is connected to terminal D on one side and jockey on the other. The position of jockey is adjusted on the wire so that balance point is obtained and galvanometer shows no deflection. Let the length of resistance of wire AB = kl. Resistance of wire BC = k (100 l).

    According to the principle of Wheatstone bridge,



 

Potentiometer:

    Potentiometer is also a device used for measuring potential difference accurately.

Principle:

    Whenever steady current passes through a wire of uniform area of cross-section, potential difference between any two points on the wire is directly proportional to the length of the wire between the points.

Proof:

    Consider a wire of resistivity r and uniform area of cross-section, therefore, resistance per unit length will be


    Resistivity of material is always constant if physical conditions remains unchanged, thus if area of cross section is also constant, we can say that resistance per unit length is constant, the potential difference per unit length is,

                    e = r ´ I    (if I is constant, e is also constant)

and the potential difference per unit length can be used to find potential difference between two points as

V = el or V µ l

    which is the potentiometer principle.

 

 

Applications of Potentiometer:

    Potentiometer can be used for three main purposes:

1.    Finding EMF of a cell:

    The circuit diagram for comparing emf is shown in figure. The auxillary circuit consists of a battery, a rehostat and a key. The cell whose emf is to be determined is connected with positive terminal at A and negative terminal connected with galvanometer. The other end of galvanometer is connected to a jockey which moves on the potentiometer wire AB.

    As positive terminal of cell is connected to A so it will have the same potential as A. Now jockey is moved on the wire, the galvanometer shows deflection. At one particular point the galvanometer shows a null deflection. This is called null point. This will be achieved if emf of a cell is equal to the potential difference between the wire AN. If length of wire AN is l, then according to potentiometer principle,

e
µ l or e = kl

    where k is the potential difference per unit length.

2.    Comparing EMF of cell:

    The circuit diagram is shown in figure (B). The method of calculating emf of comparing emf is same. The only difference is that we have to attach two cells in place of one using two keys. First the emf of one cell is determined by inserting key k1. If null point is obtained at a distance l1 from A thene1
µ l1 or e1 = k l1

    Similarly, null point is obtained for the second cell by inserting key k2. If null point is obtained l2 from A then

e2 = k l2

or


 

3.    Determination of internal resistance of cell:

    We know that internal resistance of a cell is given by


    where E, V and R are emf, potential difference and external resistance of cell respectively. The value of emf and potential difference are determined using potentiometer circuit.

    First keeping the key k open jockey is slided along the wire to get a balance point. If l1 is the length of wire giving the balance point at P, then emf of the cell is given by,

e = k l1

where k is the potential difference per unit length.

    The key is now closed and again balance point is obtained by sliding the jockey over the potentiometer wire. Let l2 be the length of wire giving the balance point in this case. This potential difference V across the terminals of the cell is,

V = k l2

    Substituting for E and V,


Primary Cell and Secondary Cell:

Primary Cell: Primary cell are those which once discharged can’t be used again or the cells in which irreversible chemical reaction takes place. For e.g. Simple voltaic cell Daniel cell, leclanche cell etc

Secondary Cell: The electrochemical cell in which reversible chemical reaction takes place. During charging electrical energy is converted to chemical energy and during discharging chemical energy gets converted into electrical energy. For e.g. lead acid accumulator, nickel cadmium battery, lithium ion battery.

Advantages and disadvantages of secondary cell:

Advantages: [a] As the reversible chemical reaction takes place, the same cell can b reused again and again after recharging.

[b] Due to smaller internal resistance of the secondary cell, even if its emf is equal to that of primary cell it can supply larger current to the circuit.

Disadvantages: [a] The initial cost of secondary cell is large as compared to the primary cell

[b] It has to be first charged before it can be put to some use.

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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.

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Capacitance

Capacitance:

    If we go on adding charge to a given body its potential goes on increasing i.e. charge on the body and potential are directly proportional to each other.

Charge µ Potential

or                             Q = CV

    where C is the constant of proportionality called the capacitance of the body. We can also define capacitance as the ratio of charge on the body to its potential. The capacitance of a capacitor depends upon:

(i)     size and shape of the conductor and nature of medium surrounding the conductor

(ii)     it also depends on the position of charges present in the neighborhood. It however is not dependent on the material of which conductor is made off.

Units of Capacitance:

    The SI unit of capacitance is Farad and capacitance is said to be one farad if

a charge of one coulomb is sufficient to raise the potential through one volt.


    The cgs unit of capacitance is statfarad and capacitance is said to be one statfarad if a charge of one e.s.u. is sufficient to raise the body potential by one statvolt.

1 Farad = 9 x 1011 stat farad

Capacity of Isolated Spherical Conductor::

    Let us consider a charge spherical body of radius r insulated from other charged bodies. If total charge on the body is q , then potential on the surface of sphere     =

    Capacity,


In cgs system,


or capacitance of a body is numerically equal to its radius.

Capacitor and Its Principle:

    A capacitor consists of two conductors separated by a certain distance with insulating medium called dielectric in between. Its main function is to increase the ability of body to take up charge. The basic principle is that the capacity of an insulated charged conductor is increased appreciably by bringing it near an earth connected uncharged conductor.

    Consider a plate A having charge +Q and potential V when another uncharged plate B is brought near this charged plate. Negative charge is induced on the inner side of this plate and positive charge is induced on the outer side of the plate B.

    The negative charge tries to decrease the potential and positive charge tends to increase it. On the whole there is net decrease in the potential of A because negative charge is near to the plate as compared with the positive charge.

On the other hand, if we connect the outer side of plate B to earth, the free positive charge on the surface disappears thereby causing a further reduction in potential. Hence as V goes on decreasing, to bring it back to the original potential we have to add lot more charge to it and thus capacity of the system increases further.

    As C = Q/V, if V decreases capacitance increases.

Parallel Plate Capacitor

:

    Parallel plate capacitor consists of parallel plates of conducting material seperated by certain fixed distance. The space in between the two plates consists of some insulating material called dielectric.

    Consider two such plates of area A with distance `d’ between them. Then the electric field between the two oppositely charged plates will be,


    Imagine a point P in between the two plates, potential difference between the two closely situated points around A is,

dV = E dr

    Potential difference between two plates,


    If instead of air there is some medium between the plates of capacitor, then


where er is the relative permittivity of the medium.

Potential Energy of Capacitor :

    Whenever a charge is added to the plates of capacitor, it increases its potential. To add more charge to it, we have to do work against coulombic repulsive force. This work done gets stored in the form of potential energy. If at any instant charge on the plates of the capacitor is q, the work done to add additional charge dq is given by,


    Total work done in increasing the charge from 0 to Q, we get,


    If V is the final potential of capacitor, then,


Force between Two Parallel Plates:

    Consider a parallel plate capacitor with distance between the plates of capacitor ‘d’. To increase this distance from d to d + Dd, the work has to be performed which is equivalent to the change in the potential energy,


    Also work done is given by,

dW = F Dd

    Equating the two values,


Substitute (ii) in (i),


Grouping of Capacitors:

Series of Grouping:

    Capacitors are said to be connected in series if second plate of first capacitor is connected with first plate of second capacitor and so on. The charge on the plates of all the capacitors is same, but potential difference will be different across different capacitors such that,

V = V1 + V2 + . . . . . . . . .+Vn


For two capacitors, C1
and C2,


Parallel Grouping:

    Capacitors are said to connected in parallel if positive plate of all the capacitors is connected to one point and negative plate to the other point. The potential difference across all the capacitors is same but the charge on the plates of capacitors is different, i.e.,

Q = Q1 + Q2 + . . . . . . . . . + Qn

Cp V = C1 V + C2 V + . . . . . . . . . + Cn V

Cp = C1 + C2 + . . . . . . . . . . + Cn

    i.e. the net capacitance is the sum of individual capacitance of all the capacitors.

Capacity of a Spherical Condenser

:

    Consider any two spherical shells of radius r1 and r2. The inner sphere is given a charge q and outer sphere is earthed. If inner sphere is given a positive charge q, there will be negative charge on the inner side of outer shell. If P be any point lying between two shells, then dV is the potential difference between two points situated a distance dr apart around P, then

dV = Edr where E = kq/r2

Potential difference V between A and B is,


When inner sphere is earthed,

    If instead of earthing outer sphere, we give a charge q to outer sphere and earth the inner sphere. If the charge induced on the inner sphere be q1, hence a charge +q1 will be present on inner surface of outer sphere, while +q2 is distributed over outer surface.

q = q1 + q2

    Thus two condensers are formed (i) between spheres A and B having capacity C1 given by 4pe0 ab/(b-a) (ii) between outer sphere and earth having capacity 4pe0b.

    Net Capacity,


Capacitance of Parallel Plate Capacitor with Dielectric Between Its Plates:

    Before finding the capacitance with dielectric between its plates we must know the behaviour of dielectric in the presence of electric field.

Polar and Non Polar Molecules :

    Polar molecules are those which are formed by the combination of two atoms having different electronegativities or we can say that molecules in which the centre of gravity of positive and negative charge do not coincide. As on the whole molecule is to be neutral therefore the magnitude of positive charge is equal to the magnitude of negative charge. Thus this system resembles a dipole and possesses a dipolemoment called its natural dipolemoment. For eg. HCl

    Non Polar Molecules are formed by joining atoms having same electronegativity. In these molecules centre of gravity of positive and negative charge coincide. Thus they do not possess dipolemoment of their own. But when non polar molecule is placed in an electric field, positive and negative charges experiences force in two opposite direction. Thus, molecules now also resembles a dipole and have dipolemoment which is called their induced dipolemoment. At some stage the electric force pulling the charges apart and the electrostatic attractive force balance each other and molecules is said to be polarised.

    Now, if a non polar dielectric slab is placed in an electric field, the atoms get polarised in the direction of . If q is the charge induced in any atom with d be distance between the two charges then total induced dipolemoment will be where is the dipolemoment and is the dipolemoment per unit volume called electric polarisation. As field acts on a dielectric a layer of positive charge is formed on the one side and a layer of negative charge on the other, this positive and negative layer generates an induced electric field . Thus net field inside the dielectric is,


Also,


    Also the ratio of applied electric field to reduced electric field

is called the dielectric constant of the medium. The polarisation is also

found to be proportional to E¢ or



    where K is called the dielectric constant of the material.

Capacitor with Dielectric Slab:

    Consider a parallel plate capacitor with plate area A and distance between capacitor plates d. Its capacitance is given by,


    Also the potential difference between the plates is given by,

    V = Ed where E is the electric field between capacitor plates. Now if we introduce a dielectric slab of thickness t between the plates of capacitor, the potential difference V¢ between plates

V¢ = E (d t) + (E Ei) t


    If instead of dielectric conducting plate is present between the plates of capacitor, then E Ei = 0 or

for conductors.


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Gauss Law

Solid Angle:

    Just as we talk about the angles in two dimensions we talk about solid angle in three dimensions. The solid angle is the measure of total opening of cone around its vertex. We measure it by drawing a sphere centered at the vertex of cone ‘a’ of the sphere intercepted by cone and dividing it by the square of distance


    The complete solid angle corresponds to the case for which ‘a’ = surface area of sphere i.e.


Electric Flux:

    Consider a closed surface S placed in uniform electric field , and divide this surface into infinitesimally small parts of each. (The surface area is a vector and its direction is same as the direction perpendicular to the plane of area)

    Electric flux is thus defined as the sum of dot products of and for all elementary areas constituting the surface. It is denoted by f and


    where q is the angle between electric field and the area vector.

Positive Electric Flux:

    If the angle between E and dS is acute then flux is said to be positive or if number of field lines leaving the surface are more than the number of electric lines of force entering the given surface.

Negative Electric Flux :

    If the number of electric lines of force entering the surface are more than the field lines leaving the surface or if the angle between E and dS is obtuse, the the flux is said to be negative flux.

GAUSS LAW:

    Electric field intensity can be calculated from Coulomb’s law for point charges only, but if we have some complex configuration of charges the field intensity can be computed using Gauss Law. It states that

    ” for any distribution of charges, the total electric flux linked with a closed surface is times the total charge within the surface “. Mathematically,


    where the first equality applies if the surface enclosed discrete charges and the second applies if the surface encloses continuous charge distribution.

Proof of Gauss’s Law:

    To prove gauss law, consider a single point charge q enclosed in a closed surface of arbitrary shape. For positive charge, the electric field is pointing radially outwards. Imagine any infinitesimally small surface such that,


But

    where dW is the solid angle subtended at O by the surface area dS.


    If there are number of charges q1, q2, …, qn
then we can write gauss law as . In cgs system, Gauss Law can be stated as .

Derivation of Coulomb’s Law from Gauss Law

:

    Consider a point charge q and we have to find the electric field at a distance ‘r‘ from charge. The gaussian surface is the spherical surface of radius ‘r‘ with centre on charge q. From symmetry electric field must have same value at all the points on the surface. Thus angle between and is zero,


    If a test charge q¢ is located at point where electric field is determined, then,


    which is nothing but Coulomb’s law.

Selecting a Gaussian Surface:

    In application of gauss’s laws to field calculations, some judgment is required in choosing the surface. Two useful guiding principles are that the point or points at which the field is to be determined must lie on the surface and the surface must have enough symmetry so that it is possible to evaluate the integral. Thus if a problem has spherical or cylindrical symmetry, the gaussian surface will usually be spherical or cylindrical respectively.

Applications of Gauss Theorem :

1.     Electric Field Due to Infinitely Long Wire of Uniform Charge Density (l) :

    In order to find electric field due to wire at P, select cylindrical surface of radius r and height l to be gaussian surface. The electric field lines are parallel to upper and lower surface of cylinder and hence makes no contribution to the electric flux. It is the curved surface which contributes to the electric flux.


    where r is the radius and l is the length of the cylinder.


2

.     Electric Field Due to Infinite Sheet of Charge with Uniform Charge Density ( s ) :

    Electric field is to be determined at P. To apply gauss theorem, let us consider a cylinder to be a gaussian surface with ends on each side of sheet as shown. Let S be the surface area of the two end surfaces. In this case field lines are parallel to curved surface, hence their contribution to flux is zero. Only end surfaces will contribute.


    Thus, we see that magnitude of electric field is independent of the distance from the sheet.

3.     Electric Field At Any Point Due to Two Charged Conducting Plates

:

    Consider two parallel plane conductors P and Q given opposite charges with s1 and s2 be the surface charge density for positive and negative conductor respectively.

    Electric field at A due to P and Q will be,


Thus, net electric field,


If s1 = s2, then, E = 0

Point lying inside two conductors:

    If point lies outside the two conductors then direction of electric field due to both the conductors is same and net electric field is,


If s1 = s2 , then E = s/e0.

4.    Electric field At Point Inside A uniformly Charged Sphere:

    Let us consider a sphere in which charge is uniformly distributed. Let r be the charge density. To find the electric field at any point inside the sphere at a distance of r from the centre. The gaussian surface is thus a sphere of radius r,


Charge inside gaussian sphere  = Charge per unit volume ´ volume

i.e                


    For point lying on the surface of the sphere,


5.     Electric Field Due To Uniformly Charged Spherical Shell :

    Consider a shell of radius R with charge uniformly distributed over its surface.

    Electric Field At Any Outside Point :

    Consider any point lying at a distance r from the centre of the shell of radius R such that ( r > R). The gaussian surface in this case is a spherical shell of radius r. At all points on this sphere electric field is same.


    According to Gauss Law,



Electric Field At A Point On the Surface

:

        In case the point P lies on the shell i.e. if r = R, the gaussian surface is assumed to be sphere of radius R. Hence E = k q/r2

Electric Field At Point Inside The Shell :

        If r < R, the electric field is zero because charge enclosed inside gaussian surface is zero.

Relation Between Surface Density Of Charge & Radius of Curvature

:

    Consider two spherical conductors with radius r1 and r2 and having charges q1 and q2. If the two spheres are connected by wire the potential will become same, i.e.

V1 = V2


    If the electric field at the two surfaces be E1 and E2, then


    Similarly, the ratio of surface charge densities on the two spheres will be,


    or surface density of charge is inversely proportional to the radius of curvature or we can say that charge always tends to concentrate towards the region with low value of radius of curvature or pointed ends.

Conductor in An Electric Field:

    In any conductor or piece of metal electrons are always acts as charge carriers and the number of electrons are always equal to the number of positive ions in it. When conductor is placed in an electric field, the positive charge carriers moves in the direction of the field and negative charge carriers in direction opposite to the field. Therefore, there is an accumulation of positive charges on one side and accumulation of negative charges on the other. These charges are called induced charges and they generate an electric field whose direction is opposite to the applied field. This electric field is called induced electric field. The accumulation of charges will keep on increasing unless applied and induced field totally balance each other. At this point, net field inside conductor becomes zero, hence force (F = qE) acting on the conductor is also reduced to zero. This whole process completes in a fraction of a second. Hence, we assume, E = 0 inside a conductor.



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Electric Potential

Potential:

    It is the term in electrostatics analogous to the temperature in heat and thermodynamics. As temperature determines the direction of flow of heat, potential determines the direction of charge flow. As heat always flow from higher to lower temperature, charge also flows from higher to lower potential (negative charge flows from lower to higher potential). The potential are classified into three categories: positive, negative and zero potential.

    Potential of a body is determined by assuming earth to be at zero potential. If we connect any body to earth and positive charge flows from the body to earth then the body is at higher potential than earth or body is at positive potential. Similarly, if charge flows from the earth to body on connecting body to earth then body is at lower potential than earth or body is at negative potential.

    Mathematically potential at a point is defined as the amount of work done in bringing a unit positive charge from infinity to that point or potential is the work done per unit charge to bring the charge from infinity to that point,


Units :

    The units of electric potential are Joules / Coulomb (Common name is volt) or ergs/statcoulomb (common name statvolt)

Relation between Volt and Stat Volt :


Potential due to Monopole

:

    Consider a charge particle Q placed at a point A. We have to find the potential at any point P which is at a distance r from Q. Now as potential is the work done in bringing unit positive charge from infinity to that point. Imagine a small displacement dx in the path of the charge. Work done to move this small distance dx will be,



    Force between charge Q and unit charge at distance x is,


    Total work done in moving from infinity to r is,


Potential At a Point Due to An Electric Dipole

Consider an electric dipole consisting of charge q and –q kept at a distance 2a from each other. The electric potential is to be calculated at point P. Let AP = r1 and BP = r2 and angle POB = q where O is the mid point of dipole. Draw BN PO and AM PO. On = OB cos q = a cos q and OM = AO cos q = a cos q.

Thus, r1 = r + a cos q and r2 = r – acos q. And the potential at P is given by

V =

V =

V =

V =

V =

But, if acos q << r, then neglecting a cosq in comparison to r, we get potential as

V =

Special Cases: [1] If point P lies on the axial line of dipole, then q = 0

V =

[2] if point P lies on the equatorial line of dipole, then q = 90

V = =0

Potential Due to a Number of Charges:

    Potential at a point due to number of charges is the sum of potential at that point due to individual charges. For e.g. if we have number of charges Q1, Q2, ……, Qn and we have to find the potential at P which is at a distance of r1, r2, ……, rn from charges Q1, Q2, ……, Qn respectively. The potential at P is,

V  = V1 + V2 + V3 + – – – – – – – -+ Vn


    The potential is a scalar quantity, so simple addition is required without considering directions.

Potential Energy:

    It is the work done in bringing number of charges from infinity to their respective positions in the absence of any other charges.

Potential Energy For A System Of Two Charges :

    It is defined as the amount of work done in bringing the charges from infinity to their respective positions in the absence of any other charge. For eg. consider two charges q1 and q2 situated a distance ‘r‘ apart. To find the potential energy of the system of charges, first find the work done in bringing q1 from infinity to A, this is equal to zero, as in absence of any other charge force acting on it is zero. Similarly we find the work done in bringing q2 from infinity to B in the presence of q1. We imagine a small path ‘dx‘ in the path at a distance ‘x‘. Work done to move this small distance is,

dW = = F dx cos180º = – Fdx

where is the force acting on q2 at P.

    To find the total work done, integrate from ¥ to r,


    This work done is equal to potential energy and hence


Potential Energy for System of Three Charges:

    Consider three charges to be situated as shown in the figure. Work done in bringing q1 in absence of any other charge is zero. Similarly as in the above article work done in bringing q2 in the presence of q1 is,


    Work done in bringing q3 in presence of q1 and q2 is,


    Thus, total work done is,


    This total work done is the potential energy of the system. Hence for a system of n-charges potential energy in generalised form is,


Potential Difference:

    It is the difference of potential between any two points and is defined as the amount of work done in moving a unit positive charge from one point to another.

    Consider two points A and B with potential VA and VB. The work done in moving a charge particle from A to B is given by,


Electric field is Conservative in Nature:

    The force or field is said to be conservative in nature if work done in moving a particle in this field around any closed path is zero or work done in moving from one point to another is independent of the path followed. For example, if unit positive charge is to be moved from A to B via path ACB, then work done will be,


    Work done in moving the particle back from B to A via some other path ACB will be


    Thus, total work done in moving a particle around any closed loop BCBCA is


    Hence as total work done is zero therefore, electric field must be conservative in nature.

Electric Potential Difference and Electric Field Intensity:

    Consider any path AB in non uniform electric field and P be any point on this curve. We know that potential difference between A and B,


    If the point A is moved to infinity, the potential at any point is given by,


    Let Q be another point situaed very close to P at a very small distance ‘dl’ so that the field between P and Q is practically the same. In that case potential difference dV between these points is


    where ET is the tangential component of electric field in direction of dl or ET = dV/dl and the term is known as potential gradient. Hence if potential is constant in a certain region of space then electric field is zero.

Equipotential Surface

:An equipotential surface is that o which potential everywhere on the surface is same.

From the defination of electric potential surface

dV =

Thus, if the potential difference between two points on the surface is zero i.e. dV =0, it implies that no work is done in moving the charge between the two points dW =0

Work done to move a charge between two points is also given by

dW = =0

As q0, E and dx are not zero this implies that cos =0 or angle between electric field intensity and displacement vector is zero. Thus, electric field intensity is always perpendicular to the equipotential surface. Thus, no two equipotential surfaces can intersect, because if they do intersect there will be two direction of electric field intensity corresponding to two surfaces at the same point which is never possible.


Note: [a] If the electric field intensity is uniform, then the equipotential surfaces are planes with their surfaces perpendicular to the electric lines of force.

[b] If we have isolated point charge positive or negative then the equipotential surfaces are concentric spheres with their center coinciding with the position of charge.

[c] equipotential surfaces are crowded in the region of strong electric fields and are far apart in the weak fields.


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Coulomb’s Law

Point Charges :

    A charged particle whose size is very small in comparison with other distances involved in the problem is called a point charge. This is a relative concept and a large body like earth can also be considered as point object.

Coulomb’s Law :

    According to this law, the force of interaction between two point charges is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. If two point charges be q1 and q2 and r is the distance between them, then,


    where k is the constant of proportionality and depends on the medium in which the charges are placed and the system of units selected.

    If the charges are placed in vacuum, in SI, , where e0 is the absolute permittivity of vacuum. Therefore,


From eq.(1),


Thus the unit of e0 will be,


and its value is found to be 8.854 ´ 10—12 C2 N—1 m—2

Therefore,


In vacuum and in cgs system k = 1.

Units of Charge :

    In S.I., the unit of charge is Coulombs.

    One Coulomb is that much charge which when placed in vacuum at a distance of 1m from an equal and similar charge would repel it with a force of 9 x 109 Newton.

    In cgs system, the unit of charge is stat coulomb or e.s.u. (electrostatic unit)

1 Coulomb = 3 x 109 stat coulomb

Another unit is e.m.u. and 1 e.m.u. of charge = 1/10 Coulomb.

Dielectric Constant or Relative Permittivity :

The force between two charges depend upon the medium between the two charges. The force between two charges q1 and q2 located at a distance r in some medium is,


where em is the absolute permittivity of that medium.

    The equation (2) gives the force between charges in vacuum,


This ratio em/e0 is called relative permittivity or dielectric constant of the medium. Also, em = e0er (From 4)

Equn. (3) gives


Important Points:

  1. Electrical force between two charged particles is very much stronger than the gravitational force between the charges, therefore when both the forces are present we can neglect gravitational force.
  2. Electric force can be both attractive as well as repulsive whereas the gravitational force is always attractive.
  3. Net electric force between charges changes with the change in medium between the charge bodies whereas the gravitational force between them is independent of the medium
  4. It’s a two body interaction i.e. the electrostatic force between the charges is independent of the presence of other charges around them
  5. The force is conservative in nature which implies that the work done in moving a point charge once around in closed loop is zero.

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Electrostatics

Electrostatics: The branch of physics which deals with the charges at rest and the resulting phenomenon. [Charge being microscopic particles can be at rest only at zero kelvin. We assume charge to be at rest in any body if they have speed but velocity is zero]

Electrodynamics; The branch of physics dealing the charges in motion is called electrodynamics.

Electric Charge:

Charge on a body or particle is the property due to which it produces and experiences electrical and magnetic effects. Some of the naturally charges particles are electron, proton etc. But a naturally uncharged body can also acquire charge through different processes.

Types Of Charge:

Charge can be of two types:

Positive Charge: it implies deficiency of electrons in the body

Negative Charge: it is excess of electrons in the body

Positive and negative are the ‘names’ given by Benjamin Franklin and has no mathematical significance.

Properties Of charge:

[a] Additive Nature of Charge: Charge is a scalar quantity and total charge on a system of bodies or charges is the vector sum of charge on individual bodies/particles.

[b] Charge Is relativistically invariant: Relativistically invariant implies that charge on the body doesn’t vary with the speed of charge particle even if the speed of charge particle is comparable to the speed of light. As mass increases with the increase in speed specific charge [i.e. q/m] decreases with increase in speed of the charge particle.

[c] Charge is conserved: In an isolated system, the total charge always remain constant. i.e. charge can be transferred from one body to another but sum of positive and negative charges in isolated system always remain constant.

[d] Charge is quantized: Quantization implies charge on any body is always integral multiple of certain minimum value ‘e’ i.e. Q= ±ne where e= 1.6×10-19C. Quantization implies that n’ can never be a rational number.

[e] Association of mass with charge: charge is always associated with mass which implies that charges body can never be massless whereas converse is not true as mass can exist without net charge on the body.

[f]Force between like charges is repulsive and force between unlike charges is attractive. But repulsion is sure test of electrification of both the bodies.

[g]Charge at rest produces electric field only, the charge in uniform motion produces both the electric and magnetic field, whereas accelerated charge produces radiating electric and magnetic field i.e. em waves]

Charging Of bodies

An object can be charged by adding or removing electrons from the body. If body is initially neutral than adding electron will make it negatively charged and removing electrons make it positively charged.

Charging of Conductors:

The methods for charging conductors are

  1. Frictional electricity or rubbing a conductor with an insulating body
  2. Charging by conduction and induction
  3. Thermionic Emission And Photoelectric Emission

Charging Of insulators:

As movement of charges is not possible inside insulators therefore insulators can’t be charged wither by conduction or induction. When insulator is rubbed with another insulator or conductor work is done against the frictional force. This work is used to transfer electrons from one body to another , the body which gains electrons becoming negatively charged and other body gets an equal positive charge.

Change in mass on electrification:

Whichever method is used to charge a body it always involves transfer of electron between two bodies. Whenever a body is negatively charged it implies the body has gained electrons therefore mass of the body will increase and whenever body is positively charged the mass of body decreases as the body has lost electrons . For every 1C of charge, the change in mass of the body is 5.68×10-12kg.