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Consider a two-dimensional solid, such as a stretched
Chapter 7, Problem 63P(choose chapter or problem)
Problem 63P
Consider a two-dimensional solid, such as a stretched drumhead or a layer of mica or graphite. Find an expression (in terms of an integral) for the thermal energy of a square chunk of this material of area A = L 2 , and evaluate the result approximately for very low and very high temperatures. Also find an expression for the heat capacity, and use a computer or a calculator to plot the heat capacity as a function of temperature. Assume that the material can only vibrate perpendicular to its own plane, i.e., that there is only one “polarization.”
Questions & Answers
QUESTION:
Problem 63P
Consider a two-dimensional solid, such as a stretched drumhead or a layer of mica or graphite. Find an expression (in terms of an integral) for the thermal energy of a square chunk of this material of area A = L 2 , and evaluate the result approximately for very low and very high temperatures. Also find an expression for the heat capacity, and use a computer or a calculator to plot the heat capacity as a function of temperature. Assume that the material can only vibrate perpendicular to its own plane, i.e., that there is only one “polarization.”
ANSWER:
Step 1 of 6
The energy 
Here, 
According to the Planck, the average energy contained in a mode at temperature T, when it is in equilibrium, is given as-
With 
For calculating the “total thermal energy” of the 2-dimensional solid,
