Download PHYS 241 Recitation

PHYS 241 Recitation
Kevin Ralphs
Week 4
Overview
• HW Questions
• Potential
• Quiz Questions
HW Questions
Ask away….
Potential
• What does it tell me?
– The change in potential energy per unit charge an
object has when moved between two points
Δ𝑈
Δ𝑉 ≡
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
𝐸𝑥
𝜕𝑉
𝜕𝑉
𝜕𝑉
=−
; 𝐸𝑦 = −
; 𝐸𝑧 = −
𝜕𝑥
𝜕𝑦
𝜕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
𝑏
Δ𝑈 = −
𝑞𝐸 ∙ 𝑑𝑙
𝑎
Potential
• The change in potential is the change in potential
energy per unit charge
𝑏
Δ𝑈
Δ𝑉 =
= − 𝐸 ∙ 𝑑𝑙
𝑞
𝑎
• For charge distributions obeying Coulomb’s law we get
the following:
𝑞𝑖
𝑉=
𝑘
𝑟𝑖
𝑖
𝑉=
𝑞
𝑑𝑞
𝑘
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
1. Direction of steepest ascent
2. How much the function is changing in that direction
𝜕
𝜕
𝜕
𝛻=
𝑥+
𝑦+ 𝑧
𝜕𝑥
𝜕𝑦
𝜕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)