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- Using \( e(\nu, T) d\nu =\frac{2\pi h}{c^2})\frac{\nu^3}{e^{\beta h \nu}-1}\, d\nu\), where \(e(\nu, T)\) is called the black-body emissivity, show that the energy radiated per unit area and time in the range \(d\lambda\) of \(\lambda\) (where \(\lambda = c/\nu \) is the wavelength) is \[ \left(\frac{2\pi c^2h}{\lambda^5}\right)(e^{\frac{\beta h c }{\lambda}}-1)^{-1} d\lambda \equiv e(\lambda, T) d\lambda.\]
- Show that the wavelength for which \(e(\lambda, T)\) is a maximum is given by \[ \beta h c = 4.965 \lambda_\text{max}\] What does \(\frac{\lambda_\text{max}\nu_{max}}{c}\) equal?
- Solar radiation has a maximum intensity near \(\lambda = 5\times 10^{-5}\)cm. Assuming that the the sun's surface is in thermal equilibrium, determine its temperature.
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