Notes: Potential (V)

A charged object in an external electric field has potential energy (PE_e).
It is the nature of the electric force that the potential energy can be written as PE_e = q V.
V has a different value at each location in space.
V units = J / C = Volts
E points from high V to low V

Relationship to E

only depends on source charges (not test charges)
V can be determined by E
Ex: constant E → ΔV = E_x Δx

Charge distributions

Point charge → V = k_e q / r
Flat uniform surface of charge → V = -2 π k_e σ d
Uniform hollow sphere → V = k_e Q / r (outside), k_e Q / R (inside)
Equipotentials for each distribution

Reminders

V is high near + charge, low near -
E points from high V to low V

Example: 2 charges (V vs. E_x)

What is the electric potential halfway between two +1.0 nC charges which are 1.0 cm apart?
What is the electric field ...?

V

V_Left = 9×10⁹ N m²/C² * 1.0×10^-9C * / .005m = 1800V
V_Right = ?
V = 3600V

E

E_Left = 9×10⁹ N m²/C² * 1.0×10^-9C * / (0.005m)² = 360000N/C to right
E_Right = ?
E = 0

+ and -

Imagine instead we had a +1.0nC charge on the left, and a -1.0nC charge on the right.
What will V be?
What will E be?

Example: Particle collision

Two electrons are fired straight at each other.
Each electron has a mass of 9.1×10^-31kg, and an initial speed of 1.0×10⁵m/s
If the electrons start 1.0m apart, what is the closest they get to each other?

NOTE

acceleration is NOT constant (draw E field of a single electron)
→ using Newton's 2nd law will not be feasible

Solution

Given: r_i = 1.0m, m = 9.1×10^-31kg, v_i = 1.0×10⁵m/s
Context: v_f = 0
Looking for r_f
Principles: Electric potential of point charge, conservation of energy, kinetic energy
V = k_e q / r, PE = q V

Pause

Will KEi be positive or negative? (ALWAYS positive)
Will PEi be positive or negative? (positive: q < 0, V < 0, PE > 0)
As electrons approach each other, will PE increase or decrease? (increase)
What must happen, then, to KE?

Solution (ctd)

KEi = 9e-21J, PEi = 2.3e-28J (ignore)
PEf = 9e-21J → rf = 2.6e-8m