Microscopic Origin of Ohm's Law
Understanding how the concepts of drift velocity and relaxation time lead to the microscopic form of Ohm's Law (J = σE or E = ρJ).
What is Microscopic Origin of Ohm's Law?
The average velocity attained by charge carriers (like electrons) in a material under the influence of an applied electric field, in a direction opposite to the field.
Key formula / rule: Drift Velocity
Key points
- Understand the concept of drift velocity and its relation to the electric field.
- Explain the role of relaxation time in determining drift velocity.
- Derive the microscopic form of Ohm's law from fundamental principles.
- Relate macroscopic quantities (current, voltage) to microscopic properties (charge density, drift velocity).
Common exam trap
Confusing drift velocity with the instantaneous velocity of electrons.
Definitions
- Term
Drift Velocity (vd)
- Meaning
The average velocity attained by charge carriers (like electrons) in a material under the influence of an applied electric field, in a direction opposite to the field.
- Term
Relaxation Time (τ)
- Meaning
The average time interval between two successive collisions of a charge carrier (e.g., an electron) with the ions in the conductor's lattice.
- Term
Current Density (J)
- Meaning
A vector quantity representing the amount of electric current flowing per unit area perpendicular to the direction of flow.
- Term
Conductivity (σ)
- Meaning
A measure of a material's ability to conduct electric current. It is the reciprocal of resistivity.
- Term
Resistivity (ρ)
- Meaning
A measure of a material's opposition to the flow of electric current. It is an intrinsic property of the material.
Learning objectives
Understand the concept of drift velocity and its relation to the electric field.
Explain the role of relaxation time in determining drift velocity.
Derive the microscopic form of Ohm's law from fundamental principles.
Relate macroscopic quantities (current, voltage) to microscopic properties (charge density, drift velocity).
Define and understand conductivity and resistivity.
Formulae
- Name
Drift Velocity
- Note
e = magnitude of electron charge, E = electric field, τ = relaxation time, m = mass of electron. The negative sign indicates direction opposite to E.
- Expression
vd = \frac{-eE\τ}{m}
- Name
Current Density
- Note
n = number density of charge carriers, q = charge of carrier (for electrons, q = -e). For electrons, J = n(-e)vd.
- Expression
J = nq vd
- Name
Microscopic Ohm's Law (Conductivity form)
- Note
σ = electrical conductivity of the material.
- Expression
J = \σ E
- Name
Conductivity
- Note
Derived from J = n(-e)vd and vd = (-eEτ)/m. Depends on material properties.
- Expression
\σ = \frac{ne2\τ}{m}
- Name
Microscopic Ohm's Law (Resistivity form)
- Note
ρ = electrical resistivity of the material.
- Expression
E = \ρ J
- Name
Resistivity
- Note
ρ = 1/σ. Depends on material properties.
- Expression
\ρ = \frac{m}{ne2\τ}
Prerequisites
Electric Field and Potential
Electric Current and Resistance
Motion of Charge Carriers
Common mistakes
Confusing drift velocity with the instantaneous velocity of electrons.
Assuming electrons move in a straight line without collisions.
Forgetting the negative sign in the drift velocity formula (though it indicates direction).
Incorrectly relating current density and electric field without considering material properties.
Keywords
Drift Velocity
Relaxation Time
Current Density
Conductivity
Resistivity
Microscopic Ohm's Law
Electric Field
Charge Carriers
Practice preview
Consider two conductors A and B made of the same material. Conductor A has twice the length and half the cross-sectional area of conductor B. If the relaxation time in conductor A is half that in conductor B, what is the…
hard
Which of the following statements correctly describes the microscopic origin of Ohm's Law?…
easy
Consider a metallic conductor. If the temperature of the conductor is increased, what is the most likely effect on its resistivity?…
medium