B.1.17—Wien’s displacement law

Black-body spectrum

A black body emits a continuous spectrum of wavelengths. The wavelength at which the emitted intensity is greatest is $$\lambda_{\max}$$.

Wien’s law

The peak wavelength and absolute temperature obey

$$\lambda_{\max}T=2.9\times10^{-3}\,\mathrm{m\,K}$$

Therefore

$$T=\frac{2.9\times10^{-3}}{\lambda_{\max}}$$

Interpret the shift

A hotter black body has a smaller peak wavelength, so its spectrum shifts toward shorter wavelengths. A cooler black body peaks at a longer wavelength.

Worked example from local Question Bank row 31596

A star's spectrum peaks at $740\,\mathrm{nm}=740\times10^{-9}\,\mathrm{m}$.

$$T=\frac{2.9\times10^{-3}}{740\times10^{-9}}=3.9\times10^3\,\mathrm{K}\approx4000\,\mathrm{K}$$

The wavelength conversion is essential because Wien's constant is in metres kelvin.

Calculation checks

Use $$\lambda_{\max}$$ in metres and T in kelvin. The law identifies the peak of the spectrum; it does not say that the object emits only that one wavelength.