Last time… - +

Because the electric field is a vector, calculating it can be painful. Why? (1) Must do three integrals (x, y, z). (2) The integrals are often not easy.

Sometime it is easier to think about energy, rather than force.

A reminder about potential energy (势能).

ΔK + (– Wby Earth) = 0 System: ball + Earth Surroundings: none h We define this to be the change in potential energy, ΔU.

ΔK + ΔU = 0 System: ball + Earth Surroundings: none h We find that ΔU = mgΔy for (object + Earth) near the surface of the Earth.

Definition of potential energy The change in potential energy of a system is the negative of the work done by forces internal to the system.

System: proton + capacitor Surroundings: none + - A B +

System: proton + capacitor Surroundings: none + - A B +

+ - A B System: proton + capacitor Surroundings: none + Potential energy goes down, kinetic energy goes up.

System: electron + capacitor Surroundings: none + - A B -

- - A B System: electron + capacitor Surroundings: none + Potential energy goes up, kinetic energy goes down.

- - A B A B + + 电势 In both cases, This part is called the potential difference ΔV. 电势 - + A B

For a uniform electric field: Units: Joules / Coulomb, or Volt The electrical potential energy of a particle with charge q is given by:

Example: We move a distance of 2 meters at an angle of 30o to a uniform electric field (100 N/C). What is the change in potential? 100 N/C 2 m

Example: A proton move a distance of 2 meters at an angle of 30o to a uniform electric field (100 N/C). What is the change in potential energy? 100 N/C 2 m +

IMPORTANT:

Potential difference does not depend on the path. Example: What if I went by the green path? Potential difference does not depend on the path. 100 N/C 2 m 1

For a non-uniform electric field, we integrate along the path: which we can write as the sum of three separate integrals:

System: Two positive charges + B + What is the potential difference between A and B?

A + B Now integrate along the path:

A + B Now integrate along the path:

Potential difference near a point charge B A +

Potential at one location Let rA go to infinity… B +

Potential at one location The potential near a point charge, relative to infinity: + +

Potential energy of two charges The potential energy of two point charges, relative to infinity: + +

Summary The potential difference between two points: The potential energy difference for a charge q, moved between two points: The potential near a point charge, with respect to infinity: