Electric Current In Conductors

• Electric current (i):

  • Scalar quantity with direction (not a vector)
  • Measured in amperes (A)
  • Represents rate of charge flow: $i = \frac{dQ}{dt}$
  • Does not add like vectors

• Current density (j):

  • True vector quantity
  • Measured in A/m²
  • Direction matches charge motion for positive charges; opposite for negative charges
  • Relation to current: $j = \frac{i}{A}$ for uniform current perpendicular to area
  • For non-perpendicular areas: $j = \frac{i}{A\cos\theta}$
  • General relationship: $i = \int \vec{j} \cdot d\vec{S}$

Relationship between drift speed and current density:

• Current density j = nevd
  • n = number of free electrons per unit volume
  • e = elementary charge (1.6 × 10⁻¹⁹ C)
  • vd = drift velocity of electrons

Influence factors:

1. Electric field (E): vd ∝ E, therefore j ∝ E (Ohm's Law: j = σE)
2. Material properties:
   • Electron density (n): More free electrons → higher current density
   • Collision time (τ): Longer time between collisions → higher drift speed
   • Temperature: Higher temperature → more lattice vibrations → shorter τ → lower drift speed

Typical values:

• Drift speeds are extremely slow (typically ~0.1 mm/s)
• Much slower than random thermal motion (~10⁶ m/s)
• Even with slow drift, current propagation is nearly instantaneous due to field effects

Application example: In a copper wire (n ≈ 8.5 × 10²² electrons/cm³) carrying 1A through 2mm² cross-section, drift speed is only about 0.04 mm/s

Collision-drift mechanism:

1. Between collisions:

   • Electric field accelerates electron: a = eE/m
   • Electron velocity increases linearly with time in field direction
   • Acceleration period ends when collision occurs

2. At collision:

   • Electron's gained directional velocity is randomized
   • Direction becomes random again
   • Process restarts from a random initial direction

3. Steady state establishment:

   • Average distance drifted between collisions: l = (1/2)(eE/m)τ²
   • Average drift velocity: vd = l/τ = (1/2)(eE/m)τ
   • Equilibrium reached when acceleration from field balances randomization from collisions

4. Physical interpretation:

   • Electron motion resembles "random walk with bias"
   • Collision frequency limits drift velocity
   • Steady drift is statistical outcome of many electrons
   • Drift velocity remains constant despite continuous acceleration/deceleration cycles

Key insight: Without collisions, electrons would continuously accelerate and no steady current would be established. Paradoxically, the resistivity created by collisions is what enables a steady electric current.

The critical temperature (Tc) can be identified as the precise temperature where:

• Resistivity abruptly and discontinuously drops from a finite value to exactly zero
• The material transitions from normal electron transport to superconducting behavior
• Cooper pairs (bound electron pairs) begin to form and condense into a quantum state
• The material begins to exhibit perfect diamagnetism (Meissner effect)

Note: The transition occurs over an extremely narrow temperature range, unlike normal metals which show continuous resistivity changes with temperature.

In a superconducting loop (below Tc):

• Current persists indefinitely because resistivity is exactly zero, not just very small
• Electrons move as Cooper pairs in a coherent quantum state, not as individual particles
• The energy gap in the electron spectrum prevents scattering events that would cause resistance
• Quantum tunneling allows the current to flow without energy dissipation

In normal metals:

• Resistivity is always positive, causing energy dissipation through electron-phonon interactions
• Individual electrons scatter from lattice ions, converting electrical energy to thermal energy
• Current decay follows I = I₀e^(-Rt/L), eventually reaching zero

This difference represents a true quantum phase transition, not just an extreme case of conventional conductivity.

To analyze a circuit with two parallel batteries connected to resistor R:

1. Apply Kirchhoff's loop law to each loop:

   • For the loop with battery 1: Ri + r₁i₁ - ℰ₁ = 0
   • For the loop with battery 2: Ri + r₂(i-i₁) - ℰ₂ = 0

2. Solve the system of equations to find:

   • Total current: i = (ℰ₁r₂ + ℰ₂r₁)/[(r₁+r₂)R + r₁r₂]
   • Current through battery 1: i₁ = (ℰ₁r₂ - ℰ₂r₁)/(r₁r₂) + i·r₂/(r₁+r₂)

3. Current distribution principle:

   • If ℰ₁ > ℰ₂ and r₁ = r₂, battery 1 supplies more current
   • If ℰ₁ = ℰ₂ and r₁ < r₂, battery 1 supplies more current
   • A battery may even receive current if its emf is significantly lower than the other

Note: The circuit behaves as if there were a single battery with emf ℰ₀ and internal resistance r₀.

The mathematical relationship is:

$r = R\left(\frac{l-l'}{l'}\right)$

Where:

• r = internal resistance of the battery (unknown)
• R = known external resistance connected across the battery
• l = first balance point length (with no external load)
• l' = second balance point length (with R connected)

This formula works because:

• At the first balance point: ℰ = (l/L)V₀
• At the second balance point: ℰR/(R+r) = (l'/L)V₀
• When these equations are combined and solved for r, we get the formula above

The ratio of lengths directly relates to the voltage drop caused by internal resistance.

Effects on charging time (where charging time is proportional to the time constant $\tau = RC$):

1. Double resistance only:

   • Time constant doubles: $\tau = 2R \cdot C = 2RC$
   • Charging rate halves
   • Time to reach any given percentage of full charge doubles

2. Double capacitance only:

   • Time constant doubles: $\tau = R \cdot 2C = 2RC$
   • Charging rate halves
   • Time to reach any given percentage of full charge doubles

3. Double both resistance and capacitance:

   • Time constant quadruples: $\tau = 2R \cdot 2C = 4RC$
   • Charging rate drops to 1/4 of original
   • Time to reach any given percentage of full charge increases by factor of 4

Note: While the maximum charge $q_{max} = \mathcal{E}C$ increases with capacitance, the time to reach any fixed percentage of the maximum always scales with $RC$.

The time constant (τ) in an RC circuit is: $\tau = RC$

After exactly one time constant has elapsed:

• 37% of the initial charge remains on the capacitor
• 63% of the initial charge has been discharged
• Mathematically: $q = Q e^{-1} = 0.37Q$

The time constant is the characteristic time scale of the discharge process. Each successive time constant period results in discharge of 63% of the remaining charge.

Thunderstorms as atmospheric batteries:

• Create charge separation: positive charge in upper portion (~6-7 km), negative in lower portion (~3-4 km)
• Generate potential differences of 20-100 MV between cloud base and Earth's surface
• Act as voltage sources opposing the normal atmospheric potential gradient

The charging/discharging cycle:

1. Fair weather regions: steady downward current (~1800 A globally) gradually neutralizes Earth's negative charge
2. Thunderstorms: replenish Earth's negative charge through lightning discharges
3. Lightning transfers ~20 C of negative charge to Earth in a single stroke
4. Cloud recharges in ~5 seconds before next possible discharge
5. Positive charge from upper thunderstorm regions migrates to the conductive layer at 50 km

This complete circuit maintains Earth's surface charge density of ~1 nanocoulomb/m² and preserves the 400 kV potential difference between Earth and upper atmosphere.