group Small Group Activity

30 min.

Vector Integrals (Contour Map)
Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

group Small Group Activity

30 min.

Number of Paths
Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

group Small Group Activity

30 min.

Work By An Electric Field (Contour Map)
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

group Small Group Activity

60 min.

The Valley
Students compute vector line integrals and explore their properties.
  • Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

group Small Group Activity

60 min.

The Wire
Students compute a vector line integral, then investigate whether this integral is path independent.

group Small Group Activity

30 min.

The Grid

group Small Group Activity

30 min.

Murder Mystery Method

None

Bottle in a Bottle 2

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 \(U=\frac32 Nk_BT\).

  1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

  2. 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).

  3. Discuss your results.

  • Found in: Energy and Entropy course(s)

group Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

face Lecture

5 min.

Energy and Entropy review
This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.