Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.

A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.

Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

See also the following more detailed problem and solution: Effective Potentials: Graphical Version

An electron is moving on a two dimension surface with a radially symmetric electrostatic potential given by the graph below:

1. Sketch the effective potential if the angular momentum is not zero.
2. Describe qualitatively, the shapes of all possible types of orbits, indicating the energy for each in your diagram.

In this unit, you will explore the electrostatic potential \(V(\vec{r})\) due to one or more discrete charges and the gravitational potential \(\Phi(\vec{r})\) due to one or more discrete masses. How does the potential vary in space? How do equipotential surfaces and the superposition principle help you answer these questions graphically? How does the value of the potential fall-off as you move away from the charges? How do power series approximations help you answer these questions algebraically?

Key Activities/Problems

• Drawing Equipotential Surfaces
• Electrostatic Potential Due to a Pair of Charges (with Series)
• Linear Quadrupole

At the end of this unit, you should be able to:

• Describe the important similarities and differences between the electrostatic potential and the gravitational potential.
• Sketch the potential due to a small number of discrete charges or masses, showing important regions of interest and qualitatively depicting the correct spacing between equipotential surfaces (or curves).
• Compute power and Laurent series expansions from a real-world problem using simple, memorized power series.
• Truncate a series properly at a given order by keeping all the terms up to that order and none of the terms of higher order.
• Discuss in detail the relationship between the graphical and algebraic representations of the potentials.

• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.

Explain why each of the statements below is true.

• Very close to either one of the charges, the equipotential curves are approximately circular.
• Very close to either one of the charges, the equipotential curves get closer together as you get closer to the charge.
• Very far from both charges, the equipotential curves are approximately circular.
• In the intermediate region where there are still distinct equipotential curves around each charge, the equipotential curves are closer together on the side away from the other charge.