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Ohm's Law, resistance|resistivity|Conductivity<






(5) Ohm's Law and Resistance


  • Ohm's law is the relation between the potential difference applied to the ends of the conductor and current flowing through the conductor.This law was expressed by George Simon Ohm in 1826
  • Statement of Ohm's Law
    'if the physical state of the conductor (Temperature and mechanical strain etc) remains unchanged ,then current flowing through a conductor is always ditectly proportional to the potential difference across the two ends of the conductor
    Mathematically
    V α I
    or
    V=IR                    (6)
    Where constant of proportionallity R is called the electric resistance or simply resistance of the conductor
  • Value of resistance depends upon the nature ,dimension and physically dimensions of the conductor
  • Ohm's Law can be deducted using drift velocity relation as given in equation -3 .Thus from the equation
    vd=(eE/m)τ
    but Now E=V/l
    Therfore
    vd=(eV/ml)τ
    Also I=neAvd
    Substituting the value of vd in I relation
    I=(ne2Aτ/ml) V                    (7)
    or V/I=(ml/ne2Aτ)=R a constant for a given conductor
    Thus
    V=IR
    Mathematical expression of Ohm's Law
    From Ohm's Law
    V=IR or R=V/I                    (8)
    Thus electric resitance is the ratio of potential difference across the two ends of conductor and amount of current flowing through the conductor
  • electric resistance of a conductor is the obstraction offered by the conductor to the flow of the current through it.
  • SI unit of resistance is ohm (Ω) where
    1 Ohm=1 volt/1 Ampere
    or 1Ω=1VA-1
  • Dimension of resistance is [ML2T-3A-2]


(6) Resistivity and conductivity


  • In terms of drift velocity ,electric current flowing through a conducting wire of length L and uniform area of cross-section A
    is
    I=dQ/dt =neAvd=(ne2Aτ/ml) V
    The above can be rearranged to give the ohm's law i.e,
    V=IR
    where R=(ml/ne2Aτ) Now R=ρl/A                    (9)
    Where ρ is called the specific resistance or resistivity of the conductor
    And ρ=m/ne2τ                    (10)
  • From equation (9) ,we can see that resistance of the wire is proportional to its length and inversly proportional to its cross-sectional area.
  • Thus resistance of a long and thin wire will greater then the resistance of short and thick wire of the same material
  • Now from equation (9)
    R=ρl/A                    (11)
    And from ohm law R=V/I
    Therefore
    ρ=(V/I)(A/L)
       =(V/L) / (I/A)
      =E/J                    (12)
    Where E=V/L is the electric field at any point inside the wire and J=I/A is current density at any point in the wire. Unit of resistivity is ohm-meter.
  • Thus from equation (12) ,electric resistivity can also be defined as the ratio of electric field intensity at any point in the conductor and the current density at that point.
  • The greater the resistivity of the material ,greater would be the field needed to establish a given current densisty
  • Perfect conductor have zero resistivities and for perfect insulators resistivity would be infinite
  • Metals and alloys have lowest resistivities and insulators have high resistivities and exceeds those of metals by a factor of 1022
  • The reciprocal of resistivity is called conductivity and is represented by σ
  • Unit of conductivity is ohm-1meter-1-1m-1) and
    σ is defined as
    σ=1/ρ
    Since ρ=E/J
    or σ=J/E
    or J=σE                    (13a)
  • The above relation can also be written in vector form as both J and E are vector quatities where vector J being directed towards E
    JE                    (13b)