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.

Students use a PhET simulation to explore the time evolution of a particle in an infinite square well potential.

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.

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.

Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

Students compute vector line integrals and explore their properties.

• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
  1. objects with like charge repel and opposite charge attract,
  2. object tend to move toward lower energy configurations
  3. The potential energy of a charged particle is related to its charge: \(U=qV\)
  4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .

Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).