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: $i = \int \vec{j} \cdot d\vec{S}$

Resistivity comparison:

• Conductors: ~10⁻⁸ Ω·m (copper: 1.72×10⁻⁸ Ω·m)
• Semiconductors: ~10⁰ Ω·m (silicon: 640 Ω·m)
• Insulators: ~10¹⁷ Ω·m (fused quartz: 7.5×10¹⁷ Ω·m)

Temperature coefficient of resistivity (α):

• Conductors: Positive (~0.004 K⁻¹), resistivity increases with temperature
• Semiconductors: Negative (~-0.05 K⁻¹), resistivity decreases with temperature
• Insulators: Varies, typically negative

Microscopic explanations:

• Conductors:

  • Increased temperature → increased lattice vibrations
  • More electron-lattice collisions → shorter collision time τ
  • Reduced drift velocity → increased resistivity

• Semiconductors:

  • Increased temperature → more thermal excitation of electrons
  • More electrons promoted from valence to conduction band
  • Increased carrier concentration → decreased resistivity
  • Thermal effects on carrier concentration dominate collision effects

• Special case - Superconductors:

  • Below critical temperature, resistivity suddenly drops to zero
  • Cooper pairs of electrons move without scattering
  • No energy dissipation during charge transport

Physical mechanisms of superconductivity:

• Below critical temperature (Tc), resistivity suddenly drops to exactly zero
• Electrons form Cooper pairs bound through lattice vibrations (phonons)
• Cooper pairs move as bosons rather than fermions
• Collective quantum state forms with macroscopic wave function
• Energy gap prevents scattering events that cause resistance

Differences from normal conductors:

• Normal conductors: Electrons scatter individually from lattice, causing resistance
• Superconductors: Cooper pairs move collectively without scattering
• Persistent currents can flow indefinitely without energy input
• Perfect diamagnetism (Meissner effect) - expulsion of magnetic fields
• Quantized magnetic flux through superconducting loops

Technological applications:

• Superconducting magnets: Extremely strong fields without heating
  • MRI machines, particle accelerators, fusion reactors
• Quantum computing: Superconducting qubits and circuits
• Maglev transportation: Frictionless train levitation
• Ultra-sensitive magnetic sensors (SQUIDs)
• Power transmission with zero loss (limited by cooling requirements)
• Ultra-fast digital switching (Josephson junctions)

Current limitations:

• Cooling requirements (highest Tc materials ~135K, still below room temperature)
• Critical current and field limitations
• Mechanical fragility of ceramic high-Tc materials
• High cost of cryogenic systems

To calculate current through an arbitrary surface:

1. Mathematical approach: Use the dot product of current density and area vectors

   • Current through the surface: i = ∫ j⃗·dS⃗

2. Importance of the angle θ between current and surface normal:

   • Only the component of current density perpendicular to the surface contributes to the net current
   • For a small area element: Δi = j·ΔS·cosθ
   • When θ = 0° (perpendicular), maximum contribution occurs (cosθ = 1)
   • When θ = 90° (parallel), no contribution occurs (cosθ = 0)

3. Physical interpretation:

   • The dot product j⃗·dS⃗ accounts for the fact that charge crossing a surface is only counted when it passes through the surface
   • Charges moving parallel to a surface don't cross it and therefore don't contribute to the current

This approach is essential for analyzing current in non-uniform fields or through irregularly shaped surfaces.

• Normal metals: Resistivity gradually increases with temperature in a continuous curve, never reaching zero
• Superconductors: Resistivity maintains a normal metal-like behavior above critical temperature (Tc), but abruptly drops to exactly zero when cooled below Tc
• This zero-resistance state below Tc allows superconductors to maintain persistent currents without energy loss
• Example: Mercury becomes superconducting at 4.2K, enabling applications like powerful electromagnets used in MRI machines and particle accelerators

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.

The terminal voltage of a battery with internal resistance decreases during discharge according to:

$V = ℰ - ir$

Where:

• V = terminal voltage (voltage measured across battery terminals)
• ℰ = electromotive force (emf) of the battery
• i = current flowing through the battery
• r = internal resistance of the battery

This relationship shows that:

• Terminal voltage is always less than the emf during discharge
• The voltage drop (ir) increases with higher current draw
• As internal resistance increases (e.g., with battery age), the voltage drop becomes more significant
• For an ideal battery where r = 0, V = ℰ

Example: A 12V battery with 0.5Ω internal resistance delivering 2A will have terminal voltage of 12V - (2A × 0.5Ω) = 11V

Parallel circuit configuration:

• All components share common electrical points at both ends
• The same voltage exists across each component
• Each component provides an independent path for current flow
• Total current is the sum of individual branch currents (i = i₁ + i₂ + i₃)

Key differences from series arrangement:

• Series: same current through all components; Parallel: current divides among paths
• Series: voltages add up to total; Parallel: same voltage across all components
• Series: total resistance increases with added components; Parallel: total resistance decreases
• Series: if one component fails open, entire circuit breaks; Parallel: if one component fails, current continues through other paths

This fundamental difference exists because parallel configurations provide multiple simultaneous paths for charge flow, while series configurations force all charge through a single path.

A Wheatstone Bridge provides superior resistance measurement accuracy due to:

1. Null measurement principle: It detects zero current rather than measuring absolute values, eliminating instrument calibration errors

2. Ratio-based measurement: The balanced equation (R₁/R₂ = R₃/R₄) depends on resistance ratios rather than absolute values, reducing systematic errors

3. Elimination of contact resistance: The measurement is independent of connection resistances that would affect direct measurements

4. Temperature coefficient compensation: When all resistors are at the same temperature, temperature effects largely cancel out

5. Sensitivity to small changes: Can detect resistance changes of 1 part in 10⁶ when properly configured

Underlying principle: When balanced, the voltage drop across R₁-R₂ path equals the voltage drop across R₃-R₄ path, creating equal potentials at the galvanometer connection points.

Example: When measuring a precision resistor of approximately 100Ω, a direct ohmmeter might have 0.5% accuracy (±0.5Ω), while a Wheatstone Bridge can achieve 0.01% accuracy (±0.01Ω).

A potentiometer with a two-way key system compares battery EMFs by:

1. The circuit contains a potentiometer wire AB, a galvanometer G, and a two-way key (S₁/S₂) connecting either Battery 1 or Battery 2
2. When S₁ is pressed, Battery 1 connects to the circuit
3. When S₂ is pressed, Battery 2 connects to the circuit
4. For each battery, the jockey is moved along the wire until finding point P where galvanometer shows zero deflection
5. The ratio of EMFs equals the ratio of balancing lengths: E₁/E₂ = l₁/l₂

No calibration is needed because only the ratio of lengths matters, not absolute values.

Example: If Battery 1 balances at 65 cm and Battery 2 at 50 cm, then E₁/E₂ = 65/50 = 1.3