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Physics of solids: Class 12 Physics
Classification of solids, crystalline and amorphous structure, mechanical properties such as stress, strain and Young's modulus, Hooke's law, elastic energy, band theory of conductors, insulators and semiconductors, and magnetic and superconducting materials.
Key points
Solids keep a fixed shape and volume because their atoms or molecules are held in position by strong forces and can only vibrate about fixed points. Solids can be crystalline, amorphous or polymeric.
Crystalline and amorphous solids:
- A crystalline solid has atoms arranged in a regular repeating pattern called a lattice. It has a sharp melting point and a definite geometric shape. Examples are salt, quartz, diamond and metals. Many show anisotropy, meaning some properties differ with direction.
- An amorphous solid has no long range order. It softens over a range of temperature and has no sharp melting point. Examples are glass, plastic and rubber. Its properties are the same in all directions (isotropic).
- A polymeric solid is made of very long chain molecules, such as polythene and rubber.
A single crystal has one continuous lattice throughout, while a polycrystalline solid is made of many tiny crystals joined together. Most metals are polycrystalline. Crystals show cleavage, which means they split along certain planes.
Deforming force: when an external force acts on a solid, its shape or size can change. Stress is the force acting per unit area, stress = F/A, in N/m² (pascal). Strain is the fractional change in size, with no unit. Longitudinal strain = ΔL/L, volumetric strain = ΔV/V and shear strain is the angle of shear.
Hooke's law: within the elastic limit, stress is directly proportional to strain. The ratio is a constant called the modulus of elasticity. Young's modulus Y = stress/strain = (F/A)/(ΔL/L) = FL/(AΔL). It measures the stiffness of a material. Its unit is N/m² (Pa). Steel has a higher Young's modulus than rubber, so it is stiffer.
A stress against strain graph for a ductile metal wire is straight up to the proportional limit. Beyond the elastic limit the wire does not return to its original length and keeps a permanent set. At the yield point the strain rises quickly with little extra stress. Then the wire is stretched until the ultimate tensile stress and finally breaks at the fracture point. Materials that break soon after the elastic limit are brittle, such as glass. Those that stretch a lot are ductile, such as copper.
Strain energy: when a wire is stretched, work is stored as elastic potential energy. Energy stored = ½ × F × ΔL. The energy per unit volume is ½ × stress × strain.
Band theory: in a single atom, electrons occupy sharp energy levels. In a solid the levels of many atoms merge into bands. The valence band holds the outer electrons and the conduction band holds free electrons. The gap between them is the forbidden gap (energy gap) Eg.
Conductors, insulators and semiconductors by bands:
- Conductor: the valence and conduction bands overlap, or the conduction band is partly filled. Many electrons are free, so conduction is easy.
- Insulator: the forbidden gap is large (about 5 eV or more), so electrons cannot reach the conduction band at ordinary temperature.
- Semiconductor: the gap is small (about 1.1 eV for silicon and 0.7 eV for germanium). At room temperature some electrons gain enough energy to cross the gap, leaving holes in the valence band. Conductivity rises as the temperature increases.
Magnetic materials: diamagnetic substances (such as bismuth and copper) are weakly repelled by a magnet. Paramagnetic substances (such as aluminium) are weakly attracted. Ferromagnetic substances (such as iron, nickel and cobalt) are strongly attracted and can become permanent magnets. They lose ferromagnetism above the Curie temperature. Hysteresis is the lag of magnetisation behind the magnetising field. Soft iron has a narrow loop, so it is used in transformer cores and electromagnets. Steel has a wide loop, so it is used for permanent magnets.
Superconductivity: some materials lose all electrical resistance when cooled below a certain critical temperature. A superconductor also expels magnetic fields from its inside (Meissner effect). Uses include powerful electromagnets in MRI machines and maglev trains.
Study tip: Learn the stress strain graph and label the proportional limit, elastic limit, yield point, ultimate stress and fracture point. For band theory questions, draw three small band diagrams and write the gap values. In numericals, convert to SI before substituting.
Definitions and flashcards
Say the meaning first, then open the card to check.
Crystal lattice
The regular three dimensional arrangement of atoms in a crystalline solid.
Amorphous solid
A solid with no regular long range order of atoms, such as glass.
Stress
The force acting per unit area of a body, in pascal.
Strain
The fractional change in length, volume or shape; it has no unit.
Young's modulus
The ratio of tensile stress to tensile strain within the elastic limit.
Elastic limit
The maximum stress up to which a body returns to its original size when released.
Forbidden gap
The energy gap between the valence and conduction bands.
Hysteresis
The lag of magnetisation behind the magnetising field in a ferromagnet.
Superconductor
A material that has zero resistance below its critical temperature.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Stress | σ = F / A | Force per unit area. |
| Longitudinal strain | ε = ΔL / L | Extension per unit original length. |
| Young's modulus | Y = (F / A) / (ΔL / L) = FL / (AΔL) | For a wire stretched by force F. |
| Elastic energy | U = ½ F ΔL | Work stored in a stretched wire. |
| Energy per unit volume | u = ½ × stress × strain | Elastic energy density. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Differentiate between crystalline and amorphous solids. (3 marks)
A crystalline solid has a regular repeating arrangement of atoms, a sharp melting point and a definite shape. Some properties depend on direction. An amorphous solid has no long range order, softens over a range of temperature and has the same properties in all directions.
State Hooke's law. (2 marks)
Within the elastic limit, the stress applied to a body is directly proportional to the strain produced in it.
Define Young's modulus and write its unit. (2 marks)
Young's modulus is the ratio of longitudinal stress to longitudinal strain within the elastic limit. Its unit is newton per square metre, or pascal.
Explain the difference between ductile and brittle materials. (2 marks)
A ductile material, such as copper, stretches a great deal and shows a large plastic region before it breaks. A brittle material, such as glass, breaks soon after the elastic limit with very little plastic deformation.
Explain conductors, insulators and semiconductors with the help of energy bands. (5 marks)
In a conductor the valence band overlaps the conduction band, so many electrons are free to move and the resistance is low. In an insulator the forbidden gap is large, about 5 eV or more, so electrons cannot jump to the conduction band at ordinary temperature and the conductivity is nearly zero. In a semiconductor the gap is small, about 1.1 eV for silicon. At absolute zero the valence band is full and the conduction band is empty, so it acts as an insulator. At room temperature some electrons gain enough thermal energy to cross the gap, leaving holes behind, so it conducts a little. More electrons cross as the temperature rises.
Why is soft iron used for transformer cores but steel for permanent magnets? (3 marks)
Soft iron has a narrow hysteresis loop, so it is easy to magnetise and demagnetise and little energy is lost in each cycle. Steel has a wide loop, so it keeps its magnetism strongly once magnetised, which is needed for a permanent magnet.
Numericals with worked steps
A steel wire of length 2.0 m and cross-sectional area 1.0 × 10⁻⁶ m² carries a load of 200 N. Young's modulus of steel is 2.0 × 10¹¹ Pa. Find the extension.
- Y = FL/(AΔL), so ΔL = FL/(AY).
- FL = 200 × 2.0 = 400.
- AY = 1.0 × 10⁻⁶ × 2.0 × 10¹¹ = 2.0 × 10⁵.
- ΔL = 400 / (2.0 × 10⁵) = 2.0 × 10⁻³ m.
Answer: ΔL = 2.0 × 10⁻³ m (2 mm)
A force of 500 N acts on a wire of cross-section 2.5 × 10⁻⁶ m². Find the stress.
- Stress = F/A.
- 500 / (2.5 × 10⁻⁶) = 200 × 10⁶.
Answer: Stress = 2.0 × 10⁸ Pa
Find the energy stored in the wire of the first question.
- U = ½ F ΔL.
- U = 0.5 × 200 × 2.0 × 10⁻³.
- 200 × 2.0 × 10⁻³ = 0.4. Half of 0.4 is 0.2.
Answer: U = 0.2 J
A wire 1.5 m long stretches by 0.75 mm. Find the strain.
- ΔL = 0.75 mm = 0.75 × 10⁻³ m.
- Strain = ΔL/L = 0.75 × 10⁻³ / 1.5 = 0.5 × 10⁻³.
Answer: Strain = 5.0 × 10⁻⁴ (no unit)
Common mistakes
- Saying amorphous solids have a sharp melting point. They soften over a range.
- Using the diameter instead of the area in Young's modulus problems. Find A = πd²/4 first.
- Forgetting to convert mm and mm² to metres and square metres.
- Writing a unit for strain. Strain is a ratio and has no unit.
- Confusing stiffness (Young's modulus) with strength (breaking stress).