*assignment*Electric Field from a Rod*assignment*Homework##### Electric Field from a Rod

Static Fields 2023 (5 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 \((0,0,2L)\).*assignment*Mass Density*assignment*Homework##### Mass Density

Static Fields 2023 (4 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.

*group*Proportional Reasoning*group*Small Group Activity10 min.

##### Proportional Reasoning

Static Fields 2023 (3 years) In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.*assignment*Gauss's Law for a Rod inside a Cube*assignment*Homework##### Gauss's Law for a Rod inside a Cube

Static Fields 2023 (4 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*Line Sources Using the Gradient*assignment*Homework##### Line Sources Using the Gradient

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

*assignment*Boltzmann probabilities*assignment*Homework##### Boltzmann probabilities

Energy Entropy Boltzmann probabilities Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies \((-\epsilon,0,\epsilon)\).- At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy \(U\)? What is the entropy \(S\)?
- At very low temperature, what are the three probabilities?
- What are the three probabilities at zero temperature? What is the internal energy \(U\)? What is the entropy \(S\)?
- What happens to the probabilities if you allow the temperature to be negative?

*assignment*Gibbs sum for a two level system*assignment*Homework##### Gibbs sum for a two level system

Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

Solve for the thermal average occupancy of the system in terms of \(\lambda\).

Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

Find an expression for the thermal average energy of the system.

Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.

*assignment*Extensive Internal Energy*assignment*Homework##### Extensive Internal Energy

Energy and Entropy 2021 (2 years)Consider a system which has an internal energy \(U\) defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal energy is an extensive quantity. What constraint does this place on the values \(\alpha\) and \(\beta\) may have?

*face*Wavelength of peak intensity*face*Lecture5 min.

##### Wavelength of peak intensity

Contemporary Challenges 2022 (3 years) This very short lecture introduces Wein's displacement law.*computer*Blackbody PhET*computer*Computer Simulation30 min.

##### Blackbody PhET

Contemporary Challenges 2021 (4 years) Students use a PhET to explore properties of the Planck distribution.-
Static Fields 2023 (6 years)
- 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.
- Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.