(i) Young modulus = tensile stress / tensile strain, E = σ/ε

The idea

The Young modulus E quantifies stiffness: how much a material stretches elastically per unit of applied stress. Tensile stress σ is the force F applied perpendicular to a specimen's cross-sectional area A, so σ = F/A, measured in pascals (Pa), where 1 Pa = 1 N m⁻². Tensile strain ε is the fractional extension: ε = ΔL/L, where ΔL is the extension and L the original length; strain is dimensionless, though it may be quoted as a percentage. Combining these gives E = σ/ε = (F/A)/(ΔL/L) = FL/(AΔL), in Pa. For example, a wire of area 1.0 × 10⁻⁶ m², original length 2.00 m, extending 1.0 mm under 50 N: σ = 5.0 × 10⁷ Pa, ε = 5.0 × 10⁻⁴, so E = 1.0 × 10¹¹ Pa. E applies only within the elastic limit, where stress is proportional to strain.

What earns the marks

• Defines tensile stress as force per unit cross-sectional area, σ = F/A, with units Pa or N m⁻².
• Defines tensile strain as extension divided by original length, ε = ΔL/L, and recognises it as dimensionless.
• States and applies E = σ/ε, equivalently E = FL/(AΔL), within the elastic limit.
• Uses consistent SI units and converts mm to m, mm² to m² and GPa to Pa correctly before substituting.

Common mistakes and how to put them right

• Using the final length instead of the original length in the strain denominator.

  Put it right: Divide extension by the original length L.

• Forgetting to convert area from mm² to m² by multiplying by 10⁻⁶.

  Put it right: Express A in m² before computing stress.

• Treating strain as having units of metres.

  Put it right: Recognise strain is a ratio of two lengths and is dimensionless.

• Applying E = σ/ε beyond the elastic limit.

  Put it right: Use the equation only where the material obeys Hooke's law.

Exam tips

• Write the defining equation, then substitute values with units shown, so method marks are visible even if arithmetic slips.
• Check unit conversions first: 1 mm² = 1 × 10⁻⁶ m² and 1 GPa = 1 × 10⁹ Pa.
• Sanity-check the magnitude: metals typically have E of order 10¹⁰–10¹¹ Pa, so an answer of 10⁵ Pa signals an error.