(modified from
K&K 4.6) We discussed in class that \begin{align}
p &= -\left(\frac{\partial F}{\partial V}\right)_T
\end{align} Use this relationship to show that
\begin{align}
p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right),
\end{align}
where \(\langle n_j\rangle\) is the number of photons
in the mode \(j\);
Solve for the relationship between pressure and internal energy.
Determine the total mass of each of the slabs below.
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation}
\rho=A\pi\sin(\pi z/h).
\end{equation}
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation}
\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)
\end{equation}
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose surface density is given by
\(\sigma=2Ah\).
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose mass density is given by
\(\rho=2Ah\,\delta(z)\).
What are the dimensions of \(A\)?
Write several sentences comparing your answers to the different cases above.
In economics, the term utility is roughly related to overall
happiness. Many things affect your happiness, including the amount of
money you have and the amount of coffee you drink. We cannot directly
measure your happiness, but we can measure how much money you
are willing to give up in order to obtain coffee or bagels. If we
assume you choose wisely, we can thus determine that your happiness
increases when you decrease your amount of money by that amount in
exchange for increasing your coffee consumption. Thus money is a
(poor) measure of happiness or utility.
Money is also a nice quantity because it is conserved---just like
energy! You may gain or lose money, but you always do so by a
transaction. (There are some exceptions to the conservation of money,
but they involve either the Fed, counterfeiters, or destruction of
cash money, and we will ignore those issues.)
In this problem, we will assume that you have bought all the coffee
and bagels you want (and no more), so that your happiness has been
maximized. Thus you are in equilibrium with the coffee shop. We will
assume further that you remain in equilibrium with the coffee shop at
all times, and that you can sell coffee and bagels back to the coffee
shop at cost.*
Thus your savings \(S\) can be considered to be a function of your
bagels \(B\) and coffee \(C\). In this problem we will also discuss the
prices \(P_B\) and \(P_C\), which you may not assume are
independent of \(B\) and \(C\).
It may help to imagine that you could possibly buy out the local supply of coffee,
and have to import it at higher costs.
The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative
relationships between your savings and the quantity of coffee and
bagels that you have. What are the units of these prices? What is
the mathematical definition of \(P_C\) and \(P_B\)?
Write down the total differential of your savings, in terms of
\(B\), \(C\), \(P_B\) and \(P_C\).
Solve for the total differential of your net worth. Your net worth
\(W\) is the sum of your total savings plus the value of the coffee and
bagels that you own. From the total differential, relate your amount of
coffee and bagels to partial derivatives of your net worth.
The internal energy is of any ideal gas can be written as
\begin{align}
U &= U(T,N)
\end{align}
meaning that the internal energy depends only on the number of
particles and the temperature, but not the volume.* The ideal gas law
\begin{align}
pV &= Nk_BT
\end{align}
defines the relationship between \(p\), \(V\) and \(T\). You may take the
number of molecules \(N\) to be constant. Consider the free adiabatic
expansion of an ideal gas to twice its volume. “Free expansion”
means that no work is done, but also that the process is also
neither quasistatic nor reversible.
What is the change in entropy of the gas? How do you know
this?
Consider the bottle in a bottle problem in a previous problem set, summarized here.
A small bottle of
helium is placed inside a large bottle, which otherwise contains
vacuum. The inner bottle contains a slow leak, so that the helium
leaks into the outer bottle. The inner bottle contains one tenth
the volume of the outer bottle, which is insulated.
The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).
A diesel
engine requires no spark plug. Rather, the air in the cylinder is
compressed so highly that the fuel ignites spontaneously when
sprayed into the cylinder.
In this problem, you may treat air as an ideal gas, which satisfies
the equation \(pV = Nk_BT\). You may also use the property of an
ideal gas that the internal energy depends only on the temperature
\(T\), i.e. the internal energy does not change for an isothermal
process. For
air at the relevant range of temperatures the heat capacity at
fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy
is given by \(U=\frac52Nk_BT\).
Note: in this problem you are expected to use only the equations
given and fundamental physics laws. Looking up the formula in a textbook
is not considered a solution at this level.
If the air is initially at room temperature
(taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\)
of the original volume, what final temperature is attained (before fuel
injection)?
Static Fields 2023 (4 years)
Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}}
+\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
Static Fields 2023 (4 years)
Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
