Which equation represents linear thermal expansion?

Which equation represents linear thermal expansion?

• A. ΔL = α ΔT
• B. ΔL = α L ΔT
• C. ΔL = β L ΔT
• D. ΔL = γ L ΔT

Answer: (B)

The concept here is that the change in length due to a temperature change is proportional to the original length and to how much the temperature changes, with α being the coefficient of linear expansion. That gives the standard formula ΔL = α L ΔT. It also means the fractional change in length is ΔL/L = α ΔT, so longer objects change more for the same temperature rise. If you drop the original length, as in ΔL = α ΔT, you’d lose the factor that makes the length change scale with how long the object is; α has units of 1/°C, so α ΔT is dimensionless, not a length, which doesn’t make physical sense for a length change. Using a different symbol like β or γ in the same structure would only be correct if those symbols were defined as the coefficient of linear expansion for that material. Without such a definition, those forms aren’t the standard expression and would be misleading. For a concrete check, take a rod with length L = 2 meters and austenitic steel-like α around 1.2×10^-5 /°C, ΔT = 30°C. Then ΔL ≈ 2 × (1.2×10^-5) × 30 ≈ 7.2×10^-4 m, or about 0.72 mm, illustrating how the length change scales with the original length and temperature change.

The concept here is that the change in length due to a temperature change is proportional to the original length and to how much the temperature changes, with α being the coefficient of linear expansion. That gives the standard formula ΔL = α L ΔT. It also means the fractional change in length is ΔL/L = α ΔT, so longer objects change more for the same temperature rise.

If you drop the original length, as in ΔL = α ΔT, you’d lose the factor that makes the length change scale with how long the object is; α has units of 1/°C, so α ΔT is dimensionless, not a length, which doesn’t make physical sense for a length change.

Using a different symbol like β or γ in the same structure would only be correct if those symbols were defined as the coefficient of linear expansion for that material. Without such a definition, those forms aren’t the standard expression and would be misleading.

For a concrete check, take a rod with length L = 2 meters and austenitic steel-like α around 1.2×10^-5 /°C, ΔT = 30°C. Then ΔL ≈ 2 × (1.2×10^-5) × 30 ≈ 7.2×10^-4 m, or about 0.72 mm, illustrating how the length change scales with the original length and temperature change.