## Line Sources Using the Gradient

• This problem is used in the following sequences
• assignment Entropy of mixing

assignment Homework

##### Entropy of mixing
Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

Suppose that a system of $N$ atoms of type $A$ is placed in diffusive contact with a system of $N$ atoms of type $B$ at the same temperature and volume.

1. Show that after diffusive equilibrium is reached the total entropy is increased by $2Nk\ln 2$. The entropy increase $2Nk\ln 2$ is known as the entropy of mixing.

2. If the atoms are identical ($A=B$), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

3. Since the Helmholtz free energy is lower for the mixed $AB$ than for the separated $A$ and $B$, it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

Note

This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

• assignment Paramagnetism

assignment Homework

##### Paramagnetism
Energy Temperature Paramagnetism Thermal and Statistical Physics 2020 Find the equilibrium value at temperature $T$ of the fractional magnetization $$\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}$$ of a system of $N$ spins each of magnetic moment $m$ in a magnetic field $B$. The spin excess is $2s$. The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where $\mu_{tot}$ is the total magnetization. Take the entropy as the logarithm of the multiplicity $g(N,s)$ as given in (1.35 in the text): $$S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}$$ for $|s|\ll N$, where $s$ is the spin excess, which is related to the magnetization by $\mu_{tot} = 2sm$. Hint: Show that in this approximation $$S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},$$ with $S_0=k_B\log g(N,0)$. Further, show that $\frac1{kT} = -\frac{U}{m^2B^2N}$, where $U$ denotes $\langle U\rangle$, the thermal average energy.
• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
• assignment Divergence Practice including Curvilinear Coordinates

assignment Homework

##### Divergence Practice including Curvilinear Coordinates

Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

1. $$\hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\hat{K} = s^2\,\hat{s}$$
7. $$\hat{L} = r^3\,\hat{\phi}$$

• assignment Curl Practice including Curvilinear Coordinates

assignment Homework

##### Curl Practice including Curvilinear Coordinates

Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

1. $$\vec{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\vec{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\vec{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\vec{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\vec{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\vec{K} = s^2\,\hat{s}$$
7. $$\vec{L} = r^3\,\hat{\phi}$$

• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
Static Fields 2023 (6 years)

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho=A\pi\sin(\pi z/h).$$
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)$$
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

assignment Homework

Static Fields 2023 (4 years)

Find the gradient of each of the following functions:

1. $$f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z}$$
2. $$\sigma(\theta,\phi)=\cos\theta \sin^2\phi$$
3. $$\rho(s,\phi,z)=(s+3z)^2\cos\phi$$

• face Chemical potential and Gibbs distribution

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.

assignment Homework

Consider the fields at a point $\vec{r}$ due to a point charge located at $\vec{r}'$.

1. Write down an expression for the electrostatic potential $V(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
2. Write down an expression for the electric field $\vec{E}(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
3. Working in rectangular coordinates, compute the gradient of $V$.
4. Write several sentences comparing your answers to the last two questions.

• assignment Volume Charge Density, Version 2

assignment Homework

##### Volume Charge Density, Version 2
charge density delta function Static Fields 2023 (6 years)

You have a charge distribution on the $x$-axis composed of two point charges: one with charge $+3q$ located at $x=-d$ and the other with charge $-q$ located at $x=+d$.

1. Sketch the charge distribution.
2. Write an expression for the volume charge density $\rho (\vec{r})$ everywhere in space.

• Static Fields 2023 (6 years)
1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: $$V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)$$