## Electric Field from a Rod

• assignment Gauss's Law for a Rod inside a Cube

assignment Homework

##### Gauss's Law for a Rod inside a Cube
Static Fields 2022 (3 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $x,y$-plane. The charge density $\lambda_0$ is constant. Find the total flux of the electric field through a closed cubical surface with sides of length $3L$ centered at the origin.
• assignment Mass Density

assignment Homework

##### Mass Density
Static Fields 2022 (3 years) Consider a rod of length $L$ lying on the $z$-axis. Find an algebraic expression for the mass density of the rod if the mass density at $z=0$ is $\lambda_0$ and at $z=L$ is $7\lambda_0$ and you know that the mass density increases
• linearly;
• like the square of the distance along the rod;
• exponentially.
• assignment Line Sources Using Coulomb's Law

assignment Homework

##### Line Sources Using Coulomb's Law
Static Fields 2022 (5 years)
1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance $s$ straight out from the midpoint, starting from Coulomb's Law.
2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
• assignment Quantum concentration

assignment Homework

##### Quantum concentration
bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side $L$; the concentration in effect is $n=L^{-3}$. Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature $kT$. (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration $n_0$ thus defined is equal to the quantum concentration $n_Q$ defined by (63): $$n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}$$ within a factor of the order of unity.
• assignment Radiation in an empty box

assignment Homework

##### Radiation in an empty box
Thermal physics Radiation Free energy Thermal and Statistical Physics 2020

As discussed in class, we can consider a black body as a large box with a small hole in it. If we treat the large box a metal cube with side length $L$ and metal walls, the frequency of each normal mode will be given by: \begin{align} \omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2} \end{align} where each of $n_x$, $n_y$, and $n_z$ will have positive integer values. This simply comes from the fact that a half wavelength must fit in the box. There is an additional quantum number for polarization, which has two possible values, but does not affect the frequency. Note that in this problem I'm using different boundary conditions from what I use in class. It is worth learning to work with either set of quantum numbers. Each normal mode is a harmonic oscillator, with energy eigenstates $E_n = n\hbar\omega$ where we will not include the zero-point energy $\frac12\hbar\omega$, since that energy cannot be extracted from the box. (See the Casimir effect for an example where the zero point energy of photon modes does have an effect.)

Note
This is a slight approximation, as the boundary conditions for light are a bit more complicated. However, for large $n$ values this gives the correct result.

1. Show that the free energy is given by \begin{align} F &= 8\pi \frac{V(kT)^4}{h^3c^3} \int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi \\ &= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3} \\ &= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3} \end{align} provided the box is big enough that $\frac{\hbar c}{LkT}\ll 1$. Note that you may end up with a slightly different dimensionless integral that numerically evaluates to the same result, which would be fine. I also do not expect you to solve this definite integral analytically, a numerical confirmation is fine. However, you must manipulate your integral until it is dimensionless and has all the dimensionful quantities removed from it!

2. Show that the entropy of this box full of photons at temperature $T$ is \begin{align} S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3 \\ &= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3 \end{align}

3. Show that the internal energy of this box full of photons at temperature $T$ is \begin{align} \frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3} \\ &= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3} \end{align}

• assignment Line Sources Using the Gradient

assignment Homework

##### Line Sources Using the Gradient

Static Fields 2022 (5 years)
1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: $$V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)$$

• assignment Cube Charge

assignment Homework

##### Cube Charge
charge density

Integration Sequence

Static Fields 2022 (5 years)
1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
2. In a new physical situation: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube.
• assignment Electric Field of a Finite Line

assignment Homework

##### Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
2. Perform the integral to find the $z$-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the $s$-component as well!)

• assignment Differential Form of Gauss's Law

assignment Homework

##### Differential Form of Gauss's Law
Static Fields 2022 (5 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.

• group Electrostatic Potential Due to a Pair of Charges (without Series)

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Pair of Charges (without Series)
Static Fields 2022 (3 years) Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.
• Static Fields 2022 (4 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $xy$-plane. The charge density $\lambda$ is constant. Find the electric field at the point $(0,0,2L)$.