4. Consider a solid of \(N\) atoms in contact with ta heat reservoir in the temperature region where the Debye \(T^{3}\) law is valid. Show that the energy fluctuation is given by
$$ \frac{\left\langle E^{2}\right\rangle-\langle E\rangle^{2}}{\langle E\rangle^{2}} \approx \frac{0.07}{N}\left(\frac{\theta_{D}}{T}\right)^{3} $$
Consider a solid of N atoms in contact with ta heat reservoir in the temperature region where the Debye T3 law is valid.
High-temperature thermal reservoir Low-temperature thermal reservoir Dr. Terror uses a heat engine like the one shown at the right. It has the following operational parameters: 1. Ty = 1430.0K 2. Ti = 336 K 3. QH = 3620.01 4. e = 10.9% (the efficiency of the engine) TA Heat engine Part A: What is the work output (W) of the engine? W J Part B: How much heat is exhausted (L) to the low-temperature reservoir? OL. Part C: What is...
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...