1 Outline and review Review on Coulomb's Law and electric field. Discussion about electric potential (energy). Coulomb’s Law in electrostatics governs the forces between two point charges:

2 Review: Coulomb’s Law While magnitude of the force is calculated using the formula in page 1. The direction of the electric force is given by: Force is a vector. The direction of the Coulomb force follows 1.The line defined by the two point charges. 2.Like charges repel, unlike charges attract.

3 Review: From Coulomb’s Law to electric field The concept of a field, here the electric field, is introduced to provide the medium for electric force of one charge to act over a distance and “instantaneously” on another charge. The definition: More often used: F = q · E Which carries a different meaning: charge in an electric field experiences a force.

4 Review: connection of Coulomb’s Law and electric field at point charge level When e-field of a point charge is concerned: The direction of the E field is defined by the direction of a positive test charge.

5 Review: add up electric fields from charges E field from multiple point charges: When the E field is generated by multiple point charges: A vector sum of E 1 to E m, here m is the number of charges. E field from charges distributed in a volume V: with

6 Review: electric field, field line, force and work Electric field line is introduced to illustrate the field:  Field lines start from positive charge, end at negative charge.  The direction of the force on a positive test charge is tangent to the field line, or follow the direction of the field line. Hence no two field lines cross.  The density represents the field strength. Electric field exerts force on a charge and moves it, doing work on the charge.  When the e-field is uniform: d UAUA UBUB AB When a charge q is moved from point A to point B by the E field, the work the electric field does to the charge is: W AB = F  d = qE  d = U A – U B = – (U B – U A ) = – ΔU i.e., the work the field does equals the negative change of the potential energy of charge q has in the field. q E

7 Review: electric field, field line, force and work Electric field exerts force on a charge and moves it, doing work on the charge.  When the e-field is NOT uniform: d UAUA UBUB AB When a charge q is moved from point A to point B by the E field where the field is a function of the location r, the work the electric field does to the charge is: The work the field does still equals the negative change of the potential energy of charge q has in the field. q E( r A )E( r B ) rArA rBrB

8 Concepts Electric potential energy and electric potential inside an electric field:  A charge inside an electric field has electric potential energy, like a mass in a gravitational field has gravitational potential energy.  When the charge is a unit positive test charge, the electric potential energy it possesses equals to the electric potential the field has at that point. Electric field is a vector field: fields from multiple charges are added together through vector addition. Electric potential is a scalar: potentials from fields of different charges are added through scalar addition. This is a lot easier than vector addition. Because electric force is conservative, the work it does to a charge inside the electric field only equals the negative change of the electric potential energy of that charge, has nothing to do with the path the charge took.

9 Electric potential from charges Electric potential from charges:  From a point charge: Q q r

10 Electric potential from charges

11 Electric potential from charges Electric potential from charges:  From multiple point charges: Q1Q1 r1r1 Q2Q2 Q3Q3 r2r2 r3r3 UAUA

12 Electric potential from charges Electric potential from charges:  From continuous charges: Charge density ρ, Volume V. q r

13 Examples Electric potential from a dipole:

14 Examples

15 Examples Electric potential from a charge disk:

16 Examples

17 Potential and potential difference Like in gravitational potential, the absolute electric potential is meaning less unless a reference point is specified.

18 Electric potential and electric field

19 example A region of space has an electric potential described by the equation V(x, y, z) = xyz. Find an expression for the electric field vector in the region.