 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)