PHYS 241 Recitation Kevin Ralphs Week 4

Overview HW Questions Potential Quiz Questions

HW Questions Ask away….

Potential What does it tell me? Why do I care? The change in potential energy per unit charge an object has when moved between two points Δ𝑉≡ Δ𝑈 𝑞 Why do I care? The energy in a system is preserved unless there is some kind of dissipative force So the potential allows you to use all the conservation of energy tools from previous courses (i.e. quick path to getting the velocity of a particle after it has moved through a potential difference)

Potential Why do I care? (cont.) If you have the potential defined over a small area, the potential function encodes the information about the electric field in the derivative 𝐸 =−𝛻𝑉 𝐸 𝑥 =− 𝜕𝑉 𝜕𝑥 ; 𝐸 𝑦 =− 𝜕𝑉 𝜕𝑦 ; 𝐸 𝑧 =− 𝜕𝑉 𝜕𝑧

Potential Word of caution: Potential is not the same as potential energy, but they are intimately related Electrostatic potential energy is not the same as potential energy of a particle. The former is the work to construct the entire configuration, while the later is the work required to bring that one particle in from infinity There is no physical meaning to a potential, only difference in potential matter. This means that you can assign any point as a reference point for the potential The potential must be continuous

Potential In a closed system with no dissipative forces Δ𝑈+𝑊=0 The work done is due to the electric force so Δ𝑈=− 𝑎 𝑏 𝑞 𝐸 ∙𝑑 𝑙

Potential The change in potential is the change in potential energy per unit charge Δ𝑉= Δ𝑈 𝑞 =− 𝑎 𝑏 𝐸 ∙𝑑 𝑙 For charge distributions obeying Coulomb’s law we get the following: 𝑉= 𝑖 𝑘 𝑞 𝑖 𝑟 𝑖 𝑉= 𝑞 𝑘 𝑑𝑞 𝑟

Potential Although vectors hold more information than scalars, special kinds of vector fields can be “compressed” into a scalar field where the change of the field in a certain direction tells you the component of the field in that direction.

Potential Gradient The gradient is a vector operator that gives two pieces of information about a scalar function Direction of steepest ascent How much the function is changing in that direction 𝛻 = 𝜕 𝜕𝑥 𝑥 + 𝜕 𝜕𝑦 𝑦 + 𝜕 𝜕𝑧 𝑧 It transforms a scalar function into a vector field where every vector is perpendicular to the function’s isolines

Potential We recover the electric field from the potential using the gradient 𝐸 =− 𝛻 𝑉 The isolines (or isosurfaces) of the potential are called equipotentials So the electric field is perpendicular to the equipotential lines (surfaces)