Chapter 17: Physics of Solids

• Types of Solids

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  • Crystalline Solids: Atoms/molecules arranged in a regular, repeating 3D pattern (e.g., NaCl, Quartz).
  • Amorphous Solids: Atoms/molecules arranged randomly, no long-range order (e.g., glass, plastic).

• Mechanical Properties of Solids

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  • Stress ($\sigma$)

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    • Internal restoring force per unit cross-sectional area.
    • $\sigma = \frac{F}{A}$.
    • Unit: Pascal (Pa) or $N/m^2$.
    • Types: Tensile, Compressive, Shear.

  • Strain ($\epsilon$)

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    • Ratio of change in dimension to original dimension.
    • Longitudinal Strain: $\epsilon_L = \frac{\Delta L}{L_o}$.
    • Volumetric Strain: $\epsilon_V = \frac{\Delta V}{V_o}$.
    • Shear Strain: $\epsilon_S = \tan\theta \approx \theta$.
    • Unit: Dimensionless.

  • Hooke's Law

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    • Within elastic limit, stress is directly proportional to strain.
    • $\sigma \propto \epsilon \implies \sigma = Y \epsilon$.
    • $Y$: Modulus of Elasticity.

  • Young's Modulus (Y)

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    • Ratio of tensile/compressive stress to longitudinal strain.
    • $Y = \frac{\text{Tensile Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L_o} = \frac{FL_o}{A\Delta L}$.
    • Unit: Pascal (Pa) or $N/m^2$.
    • Measure of stiffness/rigidity for stretching/compression.

  • Bulk Modulus (B)

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    • Ratio of volumetric stress (pressure) to volumetric strain.
    • $B = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}} = \frac{-\Delta P}{\Delta V/V_o}$.
    • Unit: Pascal (Pa).
    • Measure of resistance to compression.
    • Reciprocal is Compressibility ($K = 1/B$).

  • Shear Modulus (G) / Modulus of Rigidity

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    • Ratio of shear stress to shear strain.
    • $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\theta}$.
    • Unit: Pascal (Pa).
    • Measure of resistance to shape change.

  • Elastic Limit and Yield Strength

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    • Elastic Limit: Maximum stress a material can withstand and still return to its original shape.
    • Yield Strength: Stress at which a material begins to deform plastically (permanently).

  • Ultimate Tensile Strength and Fracture Stress

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    • Ultimate Tensile Strength (UTS): Maximum stress a material can withstand before necking begins.
    • Fracture Stress (Breaking Stress): Stress at which the material breaks.

• Electrical Properties of Solids

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  • Conductors

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    • Materials with many free electrons.
    • Low resistivity ($\sim 10^{-8}~\Omega~m$).
    • Conductivity decreases with increasing temperature (due to increased atomic vibrations hindering electron flow).
    • Examples: Copper, Aluminum, Silver.

  • Insulators

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    • Materials with very few free electrons.
    • High resistivity ($\sim 10^{10} - 10^{14}~\Omega~m$).
    • Examples: Glass, Rubber, Plastic.

  • Semiconductors

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    • Materials with resistivity between conductors and insulators ($\sim 10^{-5} - 10^6~\Omega~m$).
    • Conductivity increases with increasing temperature (more electrons gain enough energy to become free).
    • Examples: Silicon (Si), Germanium (Ge).
    • Basis for modern electronics.

  • Superconductors

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    • Materials that exhibit zero electrical resistance below a critical temperature ($T_c$).
    • No energy loss as heat.
    • Examples: Mercury ($T_c = 4.2~K$), Niobium.
    • Applications: MRI, maglev trains, power transmission (future).

• Magnetic Properties of Solids

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  • Diamagnetic Materials

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    • Weakly repelled by external magnetic fields.
    • No permanent magnetic dipole moment.
    • Relative permeability ($K_m$) slightly less than 1 ($K_m < 1$).
    • Examples: Bismuth, Copper, Water, Gold.

  • Paramagnetic Materials

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    • Weakly attracted by external magnetic fields.
    • Have permanent magnetic dipole moments, but randomly oriented.
    • Relative permeability ($K_m$) slightly greater than 1 ($K_m > 1$).
    • Examples: Aluminum, Platinum, Oxygen.

  • Ferromagnetic Materials

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    • Strongly attracted by external magnetic fields.
    • Exhibit magnetic domains (regions of aligned atomic moments).
    • Can be permanently magnetized.
    • Relative permeability ($K_m$) much greater than 1 ($K_m \gg 1$).
    • Examples: Iron, Nickel, Cobalt.
    • Curie Temperature: Temperature above which ferromagnetic materials become paramagnetic.

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