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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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