We saw previously that the spectral intensity can be expressed with respect to wavelength: \begin{align} S_{\lambda}(\lambda) &= \frac{2\pi h c^2}{\lambda^5}\frac1{e^{\frac{hc}{\lambda k_BT}}-1} \end{align} I mentioned that the peak intensity shifts to lower wavelengths at higher temperature. We can solve for the peak in the spectral intensity by taking a derivative. The resultin equation can

but the result is a non-linear equation that is a bit of a pain. So it's convenient to just have an equation. The result is known as Wien's displacement law, and states that \begin{align} \lambda_{\text{peak}} &= \frac{b}{T} \end{align} where \(b=2.9\times 10^{-3}\text{ m K}\) is called Wien's displacement constant.