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OHM's Laws

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					Chapter - 1
                                                                                                 OHM S LAW


IMPORTANT BASIC FACTS
1.   Charges (positive or negative) in motion constitute electric current.
2.   In conductors, negatively charged electrons move under the influence of applied potential.
3.   In a conductor, when electrons flow from a point A to B the conventional electric current flows from
     B to A. Thus the Electric Current flows in a direction opposite to that of the electron flow.

                                                  Electron Flor
                                       A                              B
                                                  Current Flow


4.   Charges in a conductor move whenever there is potential difference across the ends of the conductor.
5.   Negative charges move from lower to higher potential while positive charges move from higher to lower
     potential.
6.   Rate at which charges flow is the measure of the strength of current.
                      Charge          Q    ne
     i. e. Current           OR I
                       Time           t     t
     where `n' is the total number of electrons flowing in a time `t' and e is the charge on the electron.
7.   If the rate at which the charge flow varies with time, then current also varies with time. In such cases the
                                                               dq
     instantaneous current (I) is defined as I         lim        .
                                                     dt      0 dt

                                                    1 coulomb
8.   S.I. unit of current is a ampere : 1 amper
                                                       1 sec
     i.e. The current in a conductor is said to be one ampere if one coulomb of charge flows per second.
9.   One coulomb = 6.25 x 1018 Electrons

                     6.25 1018 Electron flow
     1 amper
                               sec
10. The electric current density is defined as the electric current flowing per unit area of cross section of a
    conductor when a current `I' flows through a conductor of area of cross section `A', then

                                     I . Its S.I unit is
     Electric current density J                          Am 2 .
                                     A
11. Electric current density is a vector where as electric current is a scalar.

1.    Explain briefly the microscopic view of current.
      Electric current in a conductor is due to the flow of free electrons. In the absence of a potential difference
across the ends of the conductor, the free electrons undergo random motion in all possible directions (this motion is
similar to the random motion of gas molecules). Hence there is no current flow.
       When a potential difference is applied across the ends of a conductor, an electric field is set up in the conductor.
This field exerts a force on the electrons. The negatively charged electrons drift in a direction oppoiste to the field.
This drift motion constitutes an electric current.
Note : 1. This drift motion of the electrons is not a straight path.
         2. There will be repeated collisions of the electrons with each other and also the atoms of the conductor and
            hence the resultant motion is complicated and zig-zag.
2.     Define the terms drift velocity and mobility of electrons.
      Drift velocity ( Vd ) : It is the average velocity with which free electrons get drifted towards the positive end of
the conductor under the influence of an electric field (E).

Note : Vd       E   or Vd          E , where        is a constant called the mobility of the electrons.

       Mobility ( ) : The mobility of electrons is defined as the drift velocity acquired per unit electric field strength.
                                               Vd
                                    i.e.             .
                                               E

Note : The unit of mobility is m 2V 1s 1 . Its dimension is                 ML3T 4 A 1

3.     Define relaxation time ( )
       It is the average time interval between any two successive collisions while the electrons are drifting.
Note : larger the relaxation time, lesser the number of collisions as a result less resistance and hence more current.

                                           1
                                                    I.
                                           R

4.     Derive an expression for current in terms of drift velocity.




       Consider a metallic conductor of length
to a battery to setup an electric field E in the conductor. This electric field exerts a force on the electrons as a result, the
electrons drift with a velocity Vd . Let I be the current in the conductor.
                                                                          .

     Let
conductor.
       Total volume of the conductor                                     = LA
       Total number of electrons in the entire volume                    = LAn
       Total charged carried by the electrons in the conductor Q = LAne
       If the electrons take a time
                                        L       Vd t
                                                 L
                                         t
                                                Vd

                                                total charge flow
         Electric current in the conductor =
                                                Total time taken
                                                Q
                                        I
                                                t

         substituting for both

                                        I       n A e Vd

         Since n, A and e are all constants I         Vd

       i.e., current flowing in a conductor of uniform cross sectional area is directly proportional to the drift velocity
of the electrons.

                                        Vd             I
Note : 1. Mobility of electrons
                                            E        nAeE

                                   I        n A e Vd
          2. Current density J                              n e Vd
                                   A            A

5.   State and Explain Ohm s Law. What are its limitations.
    Ohm's Law states that "At constant temperature, the current through a conductor is directly
proportional to the potential difference between its ends".
                                                            I
                             A                                                  B
                                                            V

     Consider a conductor AB. Let I be the current in a conductor and V the potential difference between its
ends A and B. Then according to Ohm's law

                             V
     I      V or V = IR or       R , a constant. Where R is called
                             I
of opposition for flow of charges.
The limitations of the Ohm s Law are:
(a) It is not applicable for electron tubes, semiconductors, discharge tubes and electrolytes. Hence are called
    non-ohmic devices. It is applicable only to metallic conductors (ohmic device)
(b) It is not applicable at very low and very high temperatures.

6.  Define resistance of a conductor and its S.I. Unit.
    The resistance of a conductor is defined as the ratio of the potential difference between the ends of the
conductor to the current through it.
      The electric current in conductors is due to the flow of electrons. During their motion, the electrons not
only collide among themselves but also with the vibrating atoms of the conductor. This collision causes
obstruction and hence it opposes the movement of electrons and is called resistance (R) of the conductor.
It is independent of V and I but depends on length, area of cross section, temperature and material of the
conductor.
     The resistance of a conductor is the opposition offered by the conductor to the flow of electric current
through it. If I be the current in a conductor and V the potential difference across its ends, then the resistance
                                                    V
of the conductor:                                R
                                                     I

                                                    1 volt
The S.I. Unit of resistance is ohm        1 ohm
                                                  1 ampere

The resistance of a conductor is said to be One Ohm if one volt of potential difference is required to
maintain a current of one ampere in it.
NOTE:
1.    To reduce current in a circuit, high resistance is used. Specially alloys are used for this purpose. These are
      Manganise, Constantan and Nichrome.
Conductance (G) :
                                                                     1
2.    The opposite of resistance is called conductance. i.e. G
                                                                    R
3.    It is the measure of the ability of given material to allow electric current to pass through it.
4.    It is the reciprocal of resistance and expressed in an unit called Siemen (S)

7.    How and on what factors does resistance of a metallic conductor depends? Hence define the term
      specific resistance of the material of the conductor.
      Resistance (R) of a metallic conductor depends on:
      (a) Nature of the material.
      (b) length of the conductor
      (c) Area of cross section of the conductor
      (d) Temperature of the material
      (e) Presence of impurities.
     At constant temperature, Resistance (R) of a metallic conductor is directly proportional to its length (L)
and inversely proportional to its area of cross section (A).
              L            L
      i.e. R     or R        where is the constant of proportionality called specific resistance or resistivity
              A           A
of the material of the conductor (A).
        RA
             if A = 1 Unit , L = 1 Unit, then =R
         L
    Thus, specific resistance ( ) of the material of the conductor is the resistance of Unit Length of the
conductor of Unit area of cross section.
      S.I. Unit of Specific resistance is Ohm-metre (m)
      It is the resistance of 1 mt. length of a conductor having 1 square metre area of cross section.
Note :
(1)    The resistivity of a conductor is a CONSTANT for a given material and it depends upon the material
       only. It is independent of length and area of cross section.
       In general, resistivity of an alloy is greater than that of its constitutent metals

         alloy    semiconductor      conductor
                                                                                                8
(2)    Amongst metallic conductors, the resistivity of silver is least 1.47 10                      m where as that of nickel is
                               8
       highest 86.84 10            m .

(3)    Electrical Conductivity            : The opposite of resistivity                                   .
                                                                                    is called conductivity.

       It is the ability of a conductor of length 1 metre and area of cross section 1 metre 2 to R
       allow current through itself.
(4)    Conductivity is the reciprocal of resistivity.                                                               t
            1
                its SI unit is Siemen/metre.

8.    Derive Ohm s law




     Consider a conductor of length
applied between its ends to setup an electrical field

      This electric field exerts a force on the electrons as a result, the electrons drift with a velocity Vd .

      Let I be the current through the conductor.
                                     I       n A e Vd ................... (1)

      where n - number of free electrons per unit volume, e - charge of an electron, Vd - drift velocity of electrons.

                                         V
                          also E           .................... (2)
                                         L
      If                                                                  E

                                             F        eE
                                     a                   ,                   where m - mass of the electron.
                                             m        m

      drift velocity Vd        a , where         is the average interval between two successive collisions and is called the
relaxation time.

                                                 eE
                                     Vd
                                                 m

      substituting for Vd in equation (1)


                                            eE               n A e2
                           I       n Ae                                  E
                                            m                  m
     substituting for E from equation (2)

                  n A e2     V
             I
                     m       L
                      mL
            V             I
                   n e2 A


                    mL
     where                       R is a constant called resistance of the conductor.
                                                                                   .
                 n e2 A

     Hence V = IR which is Ohm

                          m       L                        L
     Note : 1. R                          also R
                         ne 2     A

                                                                  m
            resistivity of the material of the conductor
                                                                 ne 2


                                                   1 ne 2
     also conductivity of the material
                                                      m

     2. J         E is another form of Ohm             .

     3. Ammeter, Voltmeter, Galvanometer           
       Diode, Triode, Transistor             
9.   What is a resistor? Write a note on colour codes of resistor.
     An object of a conducting material having a desired value of resistance is called a resistor. It is
represented by the symbol




                            First digit                                         tolerance

                                 Second digit
                                 Multiplying factor


     Resistors are usually made of carbon with a suitable binding agent molded into a cylinder enclosed in a
ceramic or plastic jacket and is provided with two leads for connection. Resistors are widely used in electronic
circuits such as those in radio, TV. etc. These resistors are commercially available.
     A colour code is used to indicate the correct value of the resistance and its percentage reliability
(tolerance). The value of the resistance is indicated by four coloured bands marked on the body of the resistor.
The first two band indicates the first two significant digits, the third ring indicates the multiplying factor and
the fourth the tolerance in percentage. If the fourth band is missing, it means the tolerance is 20%.
     The significance of different colours are given below.
      Colour                Number                  Multiplying factor       Tolerance
      Black                      0                            100
      Brown                      1                            101
      Red                        2                            102
      Orange                     3                            103
      Yellow                     4                            104
      Green                      5                            105
      Blue                       6                            106
      Violet                     7                            107
      Gray                       8                            108
      White                      9                            109
      Gold                                                    10-1                 5%
      Silver                                                  10-2                10%
      No Colour                                                                   20%

Ex: If the indicated bands are yellow, violet, red and gold the resistance value is 47 x 102 Ohm and tolerance is
 !"
     i.e.   47 x 102   !"
Note :The colour code can be easily remembered by a slogan "B B ROY of Great Britain has a Very Good
Wife with No interest in Gold or Silver."
10. Obtain an expression for the effective (equivalent) resistance of a number of resistances connected
    in series combination.
     Resistances are said to be in series if they are connected end to end such that:
     (a) Same current flows through all of them.
     (b) Potential difference across              Sum of the potential differences
         the combination                          across individual resistance

     Consider three resistance R1, R2 and R3 connected in series between the points A and B. Let a potential
difference of V be applied across the combination with the help of a battery.

                         R1                  R2        R3                         Rs
                  A                                                  B   A                B
                 I          V1               V2        V3                I

                                                                              +       —
                                 +       —
                                                                                  V
                                     V


    Let I be the current through the resistances and V1, V2 and V3 be the potential differences across R1, R2,
and R3 respectively [Fig (a)].

       The potential difference               Sum of        the potential differences
       across the combination                 across individual resistances

                                     V = V1 + V2 + V3
     By Ohm                    1
                                   = I R 1,               V 2 = I R2   and       V 3 = I R3
                V = I [R1 + R2 + R3].................(1)
     If a single resistance Rs can replace this combination such that it draws the same current I for the same
potential difference V (Fig. b) then Rs is called the effective resistance of the series combination of resistances
[ Fig (b)].
     Applying Ohm                       #s :
             V = I Rs.......................(2)
     From Equations (1) and (2)
             RS = R1 + R2 + R3
     Thus, in general the effective resistance of a number of resistances in series is equal to the sum of
their individual resistances.
Note:
1. The effective resistance of a number of resistances in series is Larger than the largest of individual
   resistances.
2.   When                                                                          #
     Rs is given by
           Rs = R + R + ............ + R                     n times
           Rs = nR
3.   Series combination is used to increase the resistance.
11. Deduce an expression for the effective resistance for a number of resistances connected in parallel
    combination.
    A number of resistances are said to be in parallel combination if each one of them is connected between
same two points such that:
      (a) The potential difference across each resistance is the same and
      (b) Total current through the combination = Sum of the currents through individual resistances.
                                                      R1
                                                                                        Rs
                                   I1
                                                                             A                  B
                                         I2           R2

                                                      R3                     I
                           I        I3                                              +       —
                                                                                        V
                                              +       —

                                                  V


     Consider three resistances R1, R2 and R3 connected in parallel between A and B. Let a common potential
difference V be applied across the combination. The main current I from the battery divides into I1, I2 and I3
along R1, R2, and R3 (Fig.a).
                                               I = I1 + I2 + I3
     By Ohm                 $

                            V                  V                       V
                   I1          , I2               and I3
                            R1                 R2                      R3

                                      1        1         1
                        I       V                           .................... (1)
                                      R1       R2        R3

     If this parallel combination be replaced by a single resistance Rp such that it draws the same current for
the same potential difference V then Rp is called effective resistance of R1, R2, and R3 in parallel.(Fig.b).

                                                               V
     Applying Ohm                              #p : I             ............. ... (2)
                                                               Rp
     Comparing equation (1) and (2) :
                   1            1         1         1
                   Rp           R1        R2        R3

    Thus, in general, the reciprocal of the effective resistance of a combination of resistances in parallel is
equal to the sum of the reciprocals of the individual resistances.
NOTE:
1.   If two resistances R1 and R2 are in parallel, then their effective resistance Rp is given by the ratio of their
     product to their sum.

            1    1           1                              R1R2
     i.e.                             or        Rp
            Rp   R1          R2                           R1      R2

2.   The effective resistance of a number of resistances in parallel is smaller than the smallest of individual
     resistances.
3.   This combinations is used to decrease the resistance.
4.   When                  %                                                              #
     resistance of the combination is

     1       1   1          1              1
                              ............. n times
     Rp      R   R          R              R

     1       n                        R
                            Rp
     Rp      R                        n

5.   When n identical resistances each of resistance R is first connected in series and then in parallel. The ratio
     of their effective resistances is

     Rs      nR
                                 Rs        n2 R
     Rp      R/n

12. Obtain an expression for branch current in a parallel combination of two resistances.
     R1 and R2 are two resistances connected in parallel between the points A and B. Let a potential difference
of V be applied across the combination by a battery.
     The main current I from the battery divides into I1 along R1 and I2 along R2.
     I1 and I2 are called branch currents.
                       I = I1 + I2 ...................................... ... (1)
     P.D. across R1 = P.D. across R2
                    I1 R 1 = I2 R 2

          I2         R1
                        Adding 1 to both sides :
          I1         R2
     I2               R1
               1                1
     I1               R2
     I2        I1      R1        R2
          I1                R2


           I         R1        R2
                                    ( from Eqn.No.1)
          I1              R2
                       IR2                               IR2
          I1                          Similarly I2
                     R1 R2                             R1 R2


     Thus, current in one branch
                                                             (Main current)             Resistance   of the other branch
                                                                                 Sum    of the resistance

NOTE: 1. When two resistances R1 and R2 are connected in parallel. Carrying currents I1 and I2

                           I1       R2
                           I2       R1

2.   When a number of resistances R1, R2, and R3 ..... Rn are all connected in parallel with a cell of EMF of E.
     carrying currents I1, I2, I3 .... In then

                                          E   E   R                    E
     I1 : I2 : I3 : ..... : In =            :   :   ................ :
                                          R1 R2 R3                     Rn

                                              main current           effective resistance of the parallel combination
3.   Current in any branch =
                                                                   resistance of that particular branch

                                                     IR p            IR p              IR p
     i. e. current in first branch I1                       , I2      R2
                                                                            ,   I3
                                                                                        R3
                                                     R1

13. Define temperature coefficient of resistance (                                   ) of metallic conductor and deduce an expression for
    the same.
     The resistance of a metallic conductor increases linearly with increase in temperature. The increase in
resistance varies from conductor to conductor and hence is a characteristic of given conductor known as
temperature coefficient of resistance ( ! ). its units are per degree centigrade in C.G.S. and per Kelvin in S.I.
     It is defined as the ratio of increase in resistance per degree centigrade rise in temperature to its
resistance at 00 C.
                                                                 Increase in resistance per degree centigrade rise in temperature
i.e. Temperature coefficient of resistance
                                                                                       Ressistance at 0 0C
     If R0 and Rt are the resistances at 00 C and t0 C respectively for a conductor, then,

                   Rt R0                        Rt      R0
              !                            !
                    t 0                              R0 t
                     R0

     Cross multiplying and rearranging Rt                    R0  1        !t
NOTE:
1. If Ro, R1 and R2 are the resistances of a conductor at 00 C, t1o C and t2oC respectively, then:
     R1 = R0 (1 + ! t1)..........................(1)
     R2 = R0 (1 + ! t2)..........................(2)
                                      R1       1 ! t1
     Eqn. (1) &      '    ()*         R2       1 ! t2
     Cross multiplying R1 + R1 !t2 = R2 + R2 !t1
     Rearranging, R1, !t2 - R2 !t1 = R2 - R1
     """"""""""""""""""""""! (R1 t2 - R2 t1) = R2 - R1
                                                 R2      R1
                                        !
                                                R1t2     R2 t1

2.   The temperature coefficient of resistance of some alloys like, manganin, constantan etc is negligibily small
     i.e. about 10 5 per celcius. Hence they are used in the manufacture of standard resistance coils.

14. Write a note on Thermistors.
     A Thermally Sensitive Resistor is called a Thermistor.
     The resistance of a thermistor varies exponentially with increase in temperature
     i.e. even for a small change in temperature there is a large variation in resistance.
     (Thermistors are made of pure semiconductors like the oxides of nickel, cobalt manganese and zinc. The
mixture of these powered oxides is heated to a suitable temperature and embedded in ceramic binders into
compact masses of required shape such as leads, rods and discs. They are provided with a pair of platinum wires
called leads for electrical connections)
      There are two types of thermistors. One type has positive temperature coefficient of resistance (PTC) - i.e.
its resistance increases exponentially with a small increase in temperature and the other type has negative
temparature coefficient of resistance (NTC) i.e its temperature decreases exponentially with small increase in
temperature - usually NTC thermistors find an extensive usage.




                                         R                   R      ae b / t




                                                                       T
     Thermistors of resistance 1 to 200 kilo ohm at 300o K are available. A small change in temperature produces
a large and sudden change in resistance (As shown in the graph)
     When a current is passed through a Thermistor power dissipated in it rises its temperature internally which
brings about the change in resistance.
     The variation of its resistance (R) with temperature (T) is given by the relation, R = ae b/T, Where a and b
are constants and T is Kelvin temperature
     Differentiating,

                dR                  b            b
                        ae b / T    2
                                            R
                dT                 T            T2
                   1 dR             b
                                        !
                   R dT            T2
     Where ! is the temperature coefficient of resistance of a thermistor and is negative.
     If R1 and R2 are the resistances of a semi-conductor at temperature T 1K and T 2K respectively, then,
                  2.303 (log R1 log R2 )
            !                            , Where T 2 is slightly higher than T 1
                          T1 T2
NOTE:
                                    1
1.   A graph of log R versus          is a straight line.
                                    T
2.   Symbol of a thermistor.                                or
USES :
1.   To measure varying temperatures and very low temperature of the order of 10 K.
2.   Used to protect electronic equipment against upsurges in current.
3.   Used in voltage regulator circuits, temperature control units, output control of amplifiers, fire alarm systems.
4.   Used to protect windings of transformers, generators and motors, also to protect T.V. picture tube.
5.   Used in Vacuum gauges, pressure gauge and altimetres.
6.   To measure the wind velocity.
15. What is a super conductor? Define the terms super conductivity and critical temperature. Briefly
    explain the same with applications.
     Electrical conductivity is defined as the ability of the substance to permit the flow of electricity through it.
     Certain metals like mercury, lead, Zinc, tin etc. when maintained at low temperature near absolute zero
(0o K) Offer no resistance to the flow of current. This phenomenon is called super conductivity first discovered
by H. Kammerling Onnes in 1911. He was awarded the Nobel prize in 1913.
     The phenomenon by which the resistivity of a conductor (substance) becomes zero below a particular
temperature is called super conductivity. The substance which exhibits this property is called a super
conductor.
     The temperature at which the resistivity becomes negligible which marks the transition of a conductor
to the super conducting stage is called critical temperature (TC) or Transition temperature.
     For Ex: Critical temp. of Mercury is 4.2o K
                               Lead     9oK
     Alloys of Niobium & tin , Niobium & zirconium - 18o K
     Currents once established in closed super conducting circuits persist for weeks without loss of energy
(battery) unless temperature is increased above the critical temperature.
     Explanation : According to the B.C.S (Bardeen, Cooper, Schriefer) theory, in super conductors, the electrons
exist in pairs with opposite spins. Known as COOPER PAIRS. The cooper pair forms a bound system and
moves in the lattice without loss or gain of energy even during collision with them.
16. Discuss the effect of magnetic field on a super conductor and hence define critical field.
     The presence of a magnetic field affects superconductivity of the materials. The superconducting quality
will be destroyed if a sufficiently strong magnetic field is applied.
     The minimum magnetic field required to just change a material from superconducting state to normal
conducting state is called critical field (BC )
a.   The strength of the critical field ( BC ) depends upon the critical temperature ( TC ) and also the material.
b.   The critical temperature decreases with increase in the magnetic field.




c.   The graph shows the variations of T with `B'. The material remains a super conductor only in the shaded
     region where both T and B are below their critical values.
     BC is maximum at T = zero.

     and TC is maximum at B = zero

d.   The variation of critical magnetic field ( BC ) with temperature (T) is approximately given by

                                   T2
         BC  T      BC  O    1
                                    2
                                   TC

17. Write a note on high temperature superconductors.
     Materials exhibit superconductivity only at very low temperatures. It is highly expensive and difficult to
maintain such low temperatures and is a serious drawback. If superconductivity is achieved at room temperature
(300K), it becomes easy to use superconductors in all electrical appliances.
     Scientists all over the world are striving hard to achieve superconductivity at high temperatures. Several
alloys of superconducting oxides have been synthesized with transition temperatures ranging from 30K to
135K. A compound Hg Ba2 Ca2 Cu3O10 has a critical temperature of 164K which is the highest reported
temperature.
    However, in conclusion ; the possible discovery of superconductors at room temperatures would make the
usage of number of super conducting devices in our daily lives common.
Application
1.   Since the resistivity of a super conductor is practically zero. Very thin wires can carry large currents and
     hence electrical equipment like fans that drive motors, refrigerators, electric trains will be much smaller in
     size.
2.   Super conducting material can be used to transmit electric power without losses is possible.
3.   Used in electron microscope, cavity resonators, frictionless magnetic bearing.
4.   Electro magnets carrying large currents are made of super conducting material.
5.   Magleves - Magnetically levitated vehicles.
6.   Super Conducting Magnets - (a) used in Nuclear Magnetic Resonance spectrometer (NMR), (b) NMR
     imaging, (c) Magnetic ore refining, (d) particle accelerator.
7.   SQUID
     a.   Super conducting quantum interference device uses super magnets for detecting very minute changes
          in the magnetic fields of the human brain to detect the disease or disorder affecting them. It consists of
          two super conductors separated by a thin insulator. This is caled Josephson Junction.
     b.   It is also used to study accurately the past and present magnetic fields of the earth.
     c.   To detect geological faults in oil wells.
NOTE:
1.   In 1986 Miller and Bednorz from Germany were awarded noble prize for finding ceramic compounds La -
     Ba - Cu -O which is super conducting at 35o K.
2.   Another oxide Y Ba2 Cu3 O7 is super conducting at 90o K
3.   Meissner effect : The permeability (Conducting power of magnetic lines of force ) of a super conductor is
     practically zero and hence it is a perfect diamagnetic substance and magnetic field can not penetrate the
     super conductor.



                                                           B 0
                                         B#0



18. Define the terms: internal resistance (r) and e.m.f. of a cell (E).
      Internal resistance of a cell is the opposition (resistance) offered by the cell (electrolyte & electrodes)
itself to the flow of charge through it.
     The Electromotive force of a cell is the amount of work done in transferring unit charge through the
circuit.
Note : The internal resistance depends upon
     (a) distance between the electrodes
     (b) the nature of the electrodes
     (c) area of the electrodes immersed in electrolyte
          EMF is not a force. It is the work done per unit charge.
          The S.I. unit of e.m.f. is volt.
    The e.m.f. of a cell is said to be 1 volt, if 1 joule of work is done in transferring 1 coulomb of charge
through the circuit.
19. Apply Ohm s Law to a circuit and derive an expression for the current.

     Consider a cell of constant e.m.f. E volt and of internal resistance r                        connected in series with an external
resistance R . Let the current through the circuit be I ampere.
                                                            E, r

                                                            +       —

                                             I
                                                                R




     By definition, the e.m.f. of a cell is the work done in driving 1 coulomb of charge round the circuit. [i.e.
through the external resistance R and through the internal resistance (r) of the cell.
                                         Work done against the                       Work done against the
         E.M.F of the cell       =                                            +
                                         external resistance R                      internal resistance r

                                 = P.D. across R                          +       p.d. across r
                                 = IR+Ir
                                                                E
                               E = I (R +r) or I                     ....................... (1)
                                                         R r

                                                 Total effective emf in the circuit
     i. e. Current through the circuit
                                                  Total resistance in the circuit

NOTE: The Potential difference between the terminals of the cell is called the terminal potential difference
and is equal to p.d. across R.

                                            ER                                            E
             Terminal P.D. V     I R                 from eqn (1) or V
                                           R r                                                r
                                                                                      I
                                                                                              R

         V     E if r   0 or R       $

     but r can never be zero

     R     $ means the cell is in an open circuit.
     Thus, the e.m.f. of a cell is the potential difference between its terminals when the cell is an open
circuit.
     The terminal P.D (V) is always less than the EMF (E) because of the internal resistance of the cell.
     When the cell is discharging,

                                                                    E V           E V
                   V    E   Ir or E         V     Ir or r                                     R
                                                                     I             V

     When cell is getting charged,

                   V    E   Ir    or E      V      IR
20. Give an experimental method of verification of Ohm s Law.


                                                   R


                                                    V
                                   A                                  Rh
                                              Ba            PK




Electrical Circuit
1. A resistance R (standard resistance) is connected in series with a battery (Ba) an ammeter (A) and a
    Rheostat (Rh) through a plug key.
2.   A voltmeter (V) is connected in parallel with the resistance.
PROCEDURE
1.    The plug key is closed and the rheostat is adjusted for a known value of current (I) in the ammeter.
2.   The corresponding Voltmeter reading (V) is noted below.
3.   The ratio V/ I is calculated.
4.   Experiment is repeated for different values of current (I) and in each case the corresponding voltmeter
     reading (V) is noted down to calculate the V/ I ratio.
5.   The results are tabulated as follows:
     Trial No.   Current I         P.D. V          R = V/ I Ohm
                 in amp          in Volts




The ratio V/ I is found to be constant within the limits of experimental errors.
Note :
1.    If a graph of V taken along X-axis and I taken along Y-axis is plotted, the resulting graph is a STRAIGHT
      LINE with a positive slope.
                                               I
2.    The slope of the straight line, tan %             conductance
                                               V
        1       1     V
                           R, the resistance
      slope   tan %   I

                                                          -oo0oo-
                                     ADDITIONAL INFORMATION
1. Effects of electric current.
    The following are some of the characteristic effects produced by an electric current.
(a) Heating Effect : When an electric current passes through a conductor heat is produced. This phenomenon
    is called heating effect of electric current.
    Consider a conductor carrying a current `I' due to a potential difference `V' maintained across its ends. The
    electrons during their motion, gain energy between two collisions and lose the same to ions, when they
    collide with them. The average energy loss per second is IV. This energy raises the temperature of the
    conductor. The heat liberated in a time `t' for which current flows is given by

         H   VIt
              I 2 Rt    V    IR by ohm' s law

    According which, the heat liberated in a conductor is directly proportional to
    (1) Square of the strength of current.
    (2) Resistance of the conductor.
    (3) Time of passage of current.
    This is known as Joule's law of heating.
                                     V 2t             V
    Note : 1. Heat liberated H                I
                                      R               R
                                                                                  V2
    2. The average power dissipated in the form of heat is P VI             I2R
                                                                                  R
       the unit of power is watt or Joule/sec.

    3. Electrical energy consumed by a device of power `P' and resistance `R' is W     P t
    4. The practical unit of electrical energy is Kilo Watt hour (1 electrical unit) - it is 1000 watt of power
                                                  J
    consumed for 1 hour i.e. 1 kwhr         103           3600 s   36   105 J .
                                                  s

(b) Magnetic Effect : When an electric current passes through a conductor, it produces a magnetic field in the
    region surrounding the conductor. This phenomenon is known as magnetic effect of electric current.
(c) Mechanical Effect : When a conductor carrying current, is kept in an external magnetic field, it experiences
    a mechanical force - This is known as mechanical effect of electric current.
(d) Chemical Effect : When an electric current is passed through some liquids, due to decomposition,
    dissocation of ions takesplace. This phenomenon is called chemical effect.
    The liquid which undergo decomposition into ions are called electrolytes and the plates through which
    current enters or leaves the electrolyte are called electrodes and this process is called electrolysis.
    The amount of substance (m) liberated or deposited at an electrode is directly proportional to
    (1) strength of the current (I) through the electrolyte.
    (2) Time of passage of current (t)
        i.e. m     It or m   zIt m    zq
        where `z' is a constant known as electro chemical equivalent (ece) of the electrolyte.
                 m    m
             z          where `q' is the charge = It.
                 It   q

      The electro chemical equivalent (z) of a substance is defined as the mass in Kg. (m) of the substance
      deposited on one of the electrodes when unit charge (q) passes through it.
      It is expressed in an unit Kg / Coulomb in S.I. system.
      If   is the density of the material deposited and A is the area of deposition, then the thickness (d) of the
                                                              m     Z It
      layer of the material deposited is given by d            A      A

(e) Optical Effect : When electric current passes through a filament bulb or a discharge tube, electron tube it
    produces light energy enabling us to sight object - This phenomenon is known as optical effect.


2.     If length of a metallic wire becomes 'n' times. Its resistance (R) becomes n2 times. i.e. Rnew                       n2 R .

                                                                                                  1                          R
3.     If radius of a metallic wire becomes 'n' times, its resistance (R) becomes                        times Rnew
                                                                                                  n4                         n4

                                                                                                                       1
4.     If area of cross section of metallic wire becomes 'n' times, its resistance (R) becomes                               times.
                                                                                                                       n2
                                                                                                                     R
                                                                                                   i.e. Rnew            .
                                                                                                                     n2
5.     Resistance of a conductor increases with decrease in density.
6.     Resistance of pure metals and metallic alloys increase with increase in temperature but resistance of a
       semiconductor decreases with increase in temperature.
7.     If 12 wires; each of resistance 'r' ohm are connected in the form of a skeleton cube, then
                                                                                           5r
       a) effective resistance between two diogonally opposite corners Reff                   Ohm
                                                                                           6
                                                                                  7r
       b) effective resistance between any two corners of the same edge Reff         Ohm
                                                                                  12
       c) If 11 wires; each of resistance 'r' ohm are connected to form a skeleton cube, then
                                                                   7r
       effective resistance across the vacant edge Reff               Ohm
                                                                   5
8.     If 'G' be the conductance;
                   1    1     1     1                                       1
       In series G                         ................   In parallel, G          G1    G2         G3     ..........
                    S   G1 G2 G3                                             P
9.     If 'P' be the power dissipated,
                 1      1    1      1                                     1
       In series P                       ................     In parallel P      P1    P2    P3    ..............
                  S     P1   P2     P3                                     P
10.    Approximate percentage in resistance = 2 x (percentage change in length by stretching)

11.    a) If a heater boils certain amoung of water in a time ' t1 ' and another heater boils the same water in a time

       ' t2 '. If t s & t p be the time taken in series and parallel to boil the same amount of water.
                                                                                                     .

       When connected in series, t s      t1     t2
                                                       t1t 2
      When connected in parallel, t p                 t1       t2
      b) The heat generated in a wire doubles when both the radius and the length of the wire is doubled.
12.   Current carriers : Charged particles whose flow in a definite direction constitute electric current.
      In conductors           electrons are charge carriers

      In electrolytes         ions are charge carriers
      In semi conductors              electrons and holes.

13.   Cell : EMF of a cell is also defined as the maximum P.D between the two electrodes of a cell when the cell
      is in open circuit. i.e. (E = V)
      (i) EMF of a cell depends upon : (a) nature of electrods (b) nature and concentration of electrolyte used
      (c) temperature of the cell.
      (ii) EMF of a cell does not depend upon : (a) size of the cell (b) quantity of electrolyte (c) distance
      between the electrodes. (d) surface area of the electrodes in contact with electrolyte.
      (iii) When no current is drawn from the cell (cell in open circuit)
                              E=V
      (iv) When current is drawn from a cell (discharging of a cell)

                              E   V       Ir        or V        E        Ir

      (energy is dissipated in the cell due to internal resisttance (r) as more and more current is drawn from the
      cell V decreases.
      (v) When a cell is being charged (i.e., current flows into the cell)
                              V   E       Ir or E          V        Ir
      (vi) Maximum power theorem : It states that output power of a source of current is maximum when the
      internal resistance of the source (r) is equal to the external resistance (R) in the circuit. i.e. R = r.
      This theorem is applicable to all types of sources of e.m.f.
      (vii) Current at maximum power ouput
                          E           E         E
                 I
                      R       r   r       r     2r
                           E2
      also pmax      I 2r
                           4r
      (When the cell is short circuited, power output is zero.)
      In this case the entire power of the battery is dissipated in the form of heat inside the cell due to internal
      resistance.
                                                                          2
                                                                    E                E2
       Power dissipated inside thebattery =                                   r
                                                                    r                 r

                                              output power
      (viii) Efficiency of a cell
                                              input power
                                               p0 VI V                                     V    IR       R
                                      &                                           also &     =        =
                                               pi     EI        E                          E I R + r   R+r
      (The efficiency of the cell is ZERO when it is short circuited)
      (ix) When the power obtained from the source is maximum, then R = r.
                              R       1
                        &                 50%
                             R+r      2
      Thus the maximum efficiency of source of emf is 50%, it means only half of the total power drawn from
      the cell is utilised for useful purpose whereas the other half is dissipated inside the cell.

      The energy dissipated inside the cell = I 2 rt .

14.   Primary cell : an arrangement in which chemical energy is converted into electrical energy due to chemical
      action takeing place in it.
      Eg :      Voltaic cell           EMF = 1.08 Volt
                Daniel cell            EMF = 1.12 Volt
                Lechlanche cell        EMF = 1.45 Volt
      Primary cell can not be recharged, but the chemical can be replaced.
      Secondary cell : A cell which the electrically energy first stored up as chemical energy (i.e. cell is
      charged). When the current is drawn, the stored chemical energy is converted into electrical energy.
      E.g Acid or lead accumulator, Alkali or Edison cell or Ni +
      Alkali cell has lower efficiency and higher resistance compared to acid accumulator.
      Secondary cell can be charged.
      Atomic Cell : Cell constructed from certain radioactive substances which emit ' -rays (electrons) over
      long periods.
15.   Fuse : A safety device used in series with electrical installation to protect the devices and the circuits.
      Fuse wire is a tin-lead alloy (63% tin + 37% lead)
      Fuse wire must have high rsistance and low melting point.
      The maximum current that can pass through the fuse wire without melting it is called 'safe current' (I).
      If 'r' be the radius of the fuse wire and "A' its area of cross section.

                I   r3 / 2   A3 / 4
      The safe current is independent of the length of the fuse wire.
                                                   -oo0oo-

				
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