Resistance, resistivity, and Ohm's law are key concepts in understanding how electric current flows through materials. These ideas help explain why some materials conduct electricity better than others and how voltage, current, and resistance are related.

Ohm's law, which states that voltage equals current times resistance, is a fundamental principle in electrical circuits. It allows us to predict how changes in voltage or resistance affect current flow, making it crucial for designing and analyzing electrical systems.

Electrical Resistance

Resistance and Resistivity

• Resistance quantifies the opposition to the flow of electric current in a material
• Measured in ohms (Ω) and represented by the symbol R
• Depends on the material's properties and dimensions (length and cross-sectional area)
• Resistivity is an intrinsic property of a material that quantifies its resistance to current flow
  • Measured in ohm-meters (Ω⋅m) and represented by the symbol ρ
  • Relates to resistance through the equation $R = \rho \frac{L}{A}$, where L is the length and A is the cross-sectional area of the material
  • Example materials with high resistivity: glass, rubber, and ceramics
  • Example materials with low resistivity: metals like copper, silver, and aluminum

Conductivity and Temperature Coefficient of Resistivity

• Conductivity is the reciprocal of resistivity and measures a material's ability to conduct electric current
  • Measured in siemens per meter (S/m) and represented by the symbol σ
  • Related to resistivity by $\sigma = \frac{1}{\rho}$
• Temperature coefficient of resistivity describes how a material's resistivity changes with temperature
  • Represented by the symbol α and measured in units of per degree Celsius (°C⁻¹)
  • For most metals, resistivity increases with increasing temperature, resulting in a positive temperature coefficient
  • Example: copper has a temperature coefficient of resistivity of approximately 0.00393 °C⁻¹, meaning its resistivity increases by about 0.393% per degree Celsius increase in temperature

Ohm's Law

Ohm's Law and Its Applications

• Ohm's law states that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, with the constant of proportionality being the resistance (R)
  • Mathematically expressed as $V = IR$
  • Can be rearranged to solve for current ($I = \frac{V}{R}$) or resistance ($R = \frac{V}{I}$)
• Applies to many materials, including most metals, over a wide range of voltages and currents
• Example: if a 10 Ω resistor has a voltage of 5 V across it, the current flowing through the resistor is $I = \frac{V}{R} = \frac{5 V}{10 Ω} = 0.5 A$

Series and Parallel Resistors

• Resistors can be connected in series or parallel to create more complex circuits
• Series resistors are connected end-to-end, and the total resistance is the sum of the individual resistances
  • For n resistors in series: $R_{total} = R_1 + R_2 + ... + R_n$
  • The current is the same through all resistors in series, while the voltage divides among them
• Parallel resistors are connected side-by-side, and the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances
  • For n resistors in parallel: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
  • The voltage is the same across all resistors in parallel, while the current divides among them

Superconductivity

Superconductivity and Its Properties

• Superconductivity is a phenomenon in which certain materials exhibit zero electrical resistance and expel magnetic fields below a characteristic critical temperature
• Superconductors have a critical temperature (Tc) below which they transition to the superconducting state
  • Example: mercury becomes a superconductor at temperatures below 4.2 K (-269°C)
• In the superconducting state, materials can carry electric current without dissipation, enabling efficient power transmission and high-field magnets
• Superconductors also exhibit the Meissner effect, where they expel magnetic fields from their interior, making them perfect diamagnets
• Applications of superconductivity include MRI machines, particle accelerators, and magnetic levitation (maglev) trains