Ohm's Law & Drift of Electrons

Table of Contents
1. What is Electric Current?
2. Unit of Electric Current
3. Electric Current in Different Materials
4. Types of Electric Current
5. Ohm's Law
6. Drift of Electrons
7. Mobility
8. Current Density
9. Solved Examples
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Think about everything you do in a day. From using your phone to turning on lights, electricity makes it all possible. The physical quantity that describes the flow of electric charge is called electric current.

In the first unit on electrostatics we studied charges at rest. Most practical applications of electricity, however, involve charges in motion. For example, an electric bulb glows when charge flows through its filament and an electric fan rotates when charge flows through its coil.

Electric Current

What is Electric Current?

The rate of flow of electric charge through any cross-section of a conductor is known as electric current. If an amount of charge ΔQ passes through a cross-section of a conductor in time interval Δt, the average current I is

I = ΔQ / Δt

For an instantaneous current, the definition is I = dq/dt. The conventional direction of electric current is the direction of motion of positive charges; it is opposite to the direction of electron motion. In metals the moving charges are electrons which carry negative charge; their motion from lower potential to higher potential constitutes an electric current in the opposite (conventional) direction.

The conductivity of a material depends on how freely electrons (or other charge carriers) can move within it.

• For current to exist there must be a net movement of charge across a surface. In the absence of an applied electric field, free electrons in a conductor move randomly and no net current flows.
• When a conductor is connected to a battery or source of potential difference an electric field is set up. Free electrons then acquire a small net drift velocity in the direction opposite to the field, producing a steady current.
• This collective directed motion of many electrons is responsible for the macroscopic quantity we call electric current.

Current is a scalar quantity. Although current has a direction (conventionally defined), it follows scalar addition (algebraic addition) rather than vector addition. The angle between wires carrying currents does not affect the scalar sum of currents in circuit analysis.

Unit of Electric Current

• Current is measured in coulombs per second.
• The SI unit of electric current is the Ampere (A).
• One ampere is the current when one coulomb of charge passes through a cross-section in one second.
• Common submultiples: milliamp (mA = 10-3 A), microamp (µA = 10-6 A), etc.
• As an illustration: if 6.241 × 10¹⁸ electrons pass a cross-section every second, the current is one ampere.

Electric Current in Different Materials

Conductors

Materials in which some electrons are only weakly bound to atoms are called conductors. These electrons, called free electrons or conduction electrons, can move under an applied electric field and carry current.

• Examples: metals (iron, silver, gold), the human body, and electrolytic solutions. Silver has the highest electrical conductivity among common metals.

Conductors

Insulators

In insulators (or dielectrics) electrons are tightly bound to atoms and there are essentially no free electrons. Under an applied electric field, electrons may shift slightly but do not move throughout the material, so no appreciable current flows.

Insulators

Semiconductors

Semiconductors behave between conductors and insulators. At low temperature they act like insulators; as temperature rises, some electrons are excited into the conduction band and can conduct. Their electrical behaviour is also strongly affected by doping, the deliberate introduction of impurities.

• When an electron in a semiconductor becomes free it leaves behind a hole that acts as a positive charge carrier. Both electrons and holes contribute to conduction.

Semiconductors

Types of Electric Current

There are two principal types of current:

• Direct Current (DC) - current flows in one direction only, e.g., current from a battery.
• Alternating Current (AC) - current changes direction periodically, e.g., mains electricity.

Representation of AC and DC Currents

Ohm's Law

Georg Simon Ohm observed that for many conductors, at constant temperature, the current through the conductor is directly proportional to the potential difference across its ends.

Ohm's Law: V ∝ I (when temperature and other physical conditions are fixed)

The proportionality constant is the resistance R of the conductor, so the law is written as

V = IR

• V is the potential difference (voltage) across the conductor.
• I is the current through the conductor.
• R is the resistance offered by the conductor; for a uniform cylindrical conductor of length ℓ and cross-sectional area A, R = ρℓ/A where ρ is the resistivity of the material.

Ohm's Law

Drift of Electrons

A conductor contains a large number of conduction electrons which move randomly because of thermal motion. In the absence of an electric field the average velocity of these electrons is zero and there is no net current.

• When an electric field is applied along a conductor, each free electron experiences an electric force opposite to the field and acquires, in addition to its random thermal velocity, a small net average velocity in a definite direction. This average velocity is called the drift velocity v_d.

Images: (a) motion of free electrons in absence of field; (b) motion in presence of an electric field.

Although individual electrons undergo frequent collisions with ions and atoms, the ensemble acquires a steady average drift velocity which produces the observed current.

Derivation of the Expression for Drift Velocity

In thermal equilibrium (no field), the average velocity of conduction electrons is zero. Under an applied field electrons accelerate between collisions and lose momentum during collisions. We introduce the average time between collisions, the relaxation time τ, to describe this process.

• Consider an electron which immediately after a collision has thermal velocity u₁ and is accelerated with acceleration a (due to the applied electric field) for a time τ₁ until the next collision. Its velocity just before the next collision is v₁ = u₁ + a τ₁.
• Taking the average over many electrons, the average thermal velocity term vanishes, leaving the average additional velocity produced by acceleration between collisions.

Hence the average (drift) velocity is

v_d = a τ

The acceleration a on an electron of charge -e and mass m in an electric field E is

a = F/m = (-eE)/m

Therefore

v_d = - (e E τ) / m

The negative sign indicates that electrons drift opposite to the direction of the electric field. The magnitude of the drift velocity is |v_d| = e E τ / m.

Typical values of τ are of the order of 10^-14 s for metals. Thus the drift velocity is very small (order 10^-4 m s^-1) even though the signal (establishment of the electric field) travels at a large fraction of the speed of light.

Relation between Drift Velocity and Electric Current

Let n be the number density of free electrons (number per unit volume) and A the cross-sectional area of a conductor. Consider electrons moving with drift speed v_d.

Number of electrons in length ℓ of the conductor = n A ℓ.

Total charge in that segment = (n A ℓ) e.

If these electrons move past a given cross-section with speed v_d, the time to pass a length ℓ is t = ℓ / v_d.

Current I is charge passing per unit time through the cross-section, so

I = n e A v_d

Mobility

Conductivity arises from mobile charge carriers. An important quantity to characterise ease of motion of carriers is the mobility µ.

• Mobility µ is defined as the magnitude of the drift velocity per unit electric field applied: µ = |v_d| / E.
• Using v_d = e E τ / m, mobility is µ = e τ / m.
• S.I. unit of mobility: m² s^-1 V^-1.
• Mobility is inversely related to the carrier mass; electrons (lighter) typically have higher mobility than holes (heavier effective mass in semiconductors).

  

Current Density

Current density J describes how current is distributed over area. It is defined as current per unit area and is a vector quantity pointing in the direction of conventional current flow.

• For a uniform current I through area A, J = I / A.
• Using I = n e A v_d, the microscopic expression is J = n e v_d, a vector in the direction of the current.
• Ohm's microscopic form: J = σ E, where σ is the electrical conductivity. Conductivity is related to mobility by σ = n e µ.
• SI unit of current density: ampere per square metre (A m^-2); dimension [L^-2 A].

• If a surface of area A is not perpendicular to the current direction but makes an angle θ with it, the effective area normal to current is A cos θ and the current through the surface is I = J A cos θ.

• If the cross-sectional area is not normal to current and makes an angle θ with the direction of current, the current through that area equals J A cos θ.
• Current density is a vector; its direction is the direction of positive charge flow (conventional current).

Solved Examples

Q1. If the instantaneous current in a metallic wire is i = (5 + 10t) A, then find the amount of charge flown through it from t = 2 s to t = 3 s.

Solution:

Q2. The figure below shows a plot of current I through the cross-section of a wire over a time interval of 10 s. Find the amount of charge that flows through the wire during this period.

Solution:

Q3. If n = 8.5 × 10²⁸ m^-3, how long does an electron take to drift from one end of a 3 m long wire to its other end? The area of the cross-section of the wire is 2.0 × 10^-6 m² and it carries a current of 3.0 A. (JEE Mains 2019)

Solution:

Q4. What is the drift velocity for the electrons in a conductor when an electric field of strength 200 V m^-1 is applied to it and the mobility of electrons is 4.5 × 10^-6 m² s^-1 V^-1?

Solution:

Q5. A current of 5 A passes through a copper conductor (resistivity = 1.7 × 10^-8 Ω m) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is 1.1 × 10^-3 m s^-1.

(JEE Mains, 2019)

Solution:

Use I = n e A v_d and σ = 1/ρ = n e µ, combine expressions to obtain µ. (Full algebraic steps and numerical evaluation omitted here; students should substitute A = π r² with r = 5 mm and use given I, v_d, ρ to solve for µ.)

Q6. An electron beam has an aperture of 1.0 mm². A total of 6 × 10¹⁶ electrons flow through any perpendicular cross-section per second. Calculate (i) the current (ii) the current density in the electron beam.

Q7. If the mean free time between collisions for electrons in copper is 2.5 × 10^-14 s, calculate their mobility (m_e = 9.1 × 10^-31 kg).

Ans:

Q8. A circuit has a battery voltage of 20 V. A lamp with a resistance of 5 Ω is connected to the circuit. Calculate the current and the power of the circuit.

Ans: