assignment Homework

Gibbs free energy

thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}

1. Find the total differential of \(G\). As always, show your work.
2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

assignment Homework

General Solution of the Harmonic Oscillator Equation, version 2

Central Forces 2023

Give the general solution of the differential equation: \[\frac{d^2 u}{d\phi^2}+u=0\]

It is NOT necessary to show any work.

assignment Homework

General Solution of the Harmonic Oscillator Equation, version 1

Central Forces 2023

Give the general solution of the differential equation: \[\frac{d^2 \Phi}{d\phi^2}+7\Phi=0\]

It is NOT necessary to show any work.

assignment Homework

Inhomogeneous Linear Equations with Constant Coefficients

Oscillations and Waves 2023 (2 years)

Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see: The Method for Inhomogeneous Equations or your differential equations text.

For the following inhomogeneous linear equation with constant coefficients, find the general solution for \(y(x)\).

\[y''+2y'-y=\sin{x} +\cos{2x}\]

assignment Homework

Differentials of Two Variables

Static Fields 2023 (8 years) Find the total differential of the following functions:

1. \(y=3u^2 + 4\cos 3v\)
2. \(y=3uv\)
3. \(y=3u^2\cos wv\)
4. \(y=u\cos(3v^2-2)\)

assignment Homework

Coffees and Bagels and Net Worth

Energy and Entropy 2021 (2 years)

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.

1. 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 \(P_B\)?

2. Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).

3. 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.

assignment Homework

Differentials of One Variable

Static Fields 2023 (6 years) Find the total differential of the following functions:

1. \(y=3x^2 + 4\cos 2x\)
2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
3. \(y=\frac{\cos 7x}{x^2}\)
4. \(y=\cos(3x^2-2)\)

assignment Homework

Zapping With d 1

Energy and Entropy 2021 (2 years)

Find the differential of each of the following expressions; zap each of the following with \(d\):

1. \[f=3x-5z^2+2xy\]

2. \[g=\frac{c^{1/2}b}{a^2}\]

3. \[h=\sin^2(\omega t)\]

4. \[j=a^x\]

5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

assignment Homework

Differential Form of Gauss's Law

Static Fields 2023 (6 years)

For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

assignment Homework

Homogeneous Linear ODE's with Constant Coefficients

ODEs math bits Oscillations and Waves 2023 (2 years)

Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:

Constant Coefficients, Homogeneous

or your differential equations text.

Answer the following questions for each differential equation below:

• identify the order of the equation,
• find the number of linearly independent solutions,
• find an appropriate set of linearly independent solutions, and
• find the general solution.

Each equation has different notations so that you can become familiar with some common notations.

1. \(\ddot{x}-\dot{x}-6x=0\)
2. \(y^{\prime\prime\prime}-3y^{\prime\prime}+3y^{\prime}-y=0\)
3. \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)

assignment Homework

Using Gibbs Free Energy

thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

1. Compute the entropy.

2. Work out the heat capacity at constant pressure \(C_p\).

3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

4. Compute the internal energy \(U\).

assignment Homework

General Solution of the Harmonic Oscillator Equation

Central Forces 2023

Give the general solution of the differential equation: \[\frac{d^2 y}{dx^2}+Ay=0\] Make sure that you can give the solution of this equation regardless of the geometric names of the dependent and independent variables and for either sign for the constant \(A\).

It is NOT necessary to show any work. You may NOT, however, give a solution that has a negative number inside a square root. I am testing whether you can recognize this equation and remember its solution. This equation comes up over and over again in physics, but disguised by different symbols. I am also testing whether you recognize that the geometric character of the equation changes depending on the sign of \(A\).

assignment Homework

Paramagnet (multiple solutions)

Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}

1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

   \[\left(\frac{\partial M}{\partial B}\right)_S \]

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?