Chapter 2. Coulomb’s law

Coulomb’s law This law is discovered by Coulomb experimentally. In the free space, the force between two point charges is proportional to the charges of them, and is inversely proportional to the square of the distance between those charges. k is the proportionality constant., whose value is determined experimentally to be If q1, q2 have the same polarity, the force is repulsive. +q2 (Permittivity of vacuum) +q1 Coulomb’s law only states that the force between two charge is related to the distance between them and their charges. It does not tells us how the interaction occurs. O

Coulomb’s torsion balance +q1 +q2 +q1 +q2

Electric field Media that transfer electric forces. Charges that fall apart interact via electric field. +q2 The electric field is generated by the charge Q and spread into the universe. The speed of electric field transmission is the same as the speed of light. The force on q2 is proportional to the electric field near the charge q2.

Electric field due to known charge distributions E-field by point charges Continuous distributions Volume charge Quantities with prime ( ′ ) symbols are associated with sources. Un-primed quantities are related with observation positions. Surface charge Line charge

Charges in a material If an electric field problem contains a physical media, it is difficult to predict electric field in the space due to the charges contained on it. If the positions of the charges are unknown, Coulomb’s law cannot be applied. molecule Molecules in a solid are aligned in the direction of the external electric field.

Electrons in an isolated atom Electron energy level 1 atom - + - - - Tightly bound electron - - - - - More freely moving electron Energy levels and the radii of the electron orbit are quantized and have discrete values. For each energy level, two electrons are accommodated at most.

Electrons in a solid Atoms in a solid are arranged in a lattice structure. The electrons are attracted by the nuclei. The amount of attractions differs for various material. Freely moving electron + - Electron energy level External E-field - Tightly bound electron To accommodate lots of electrons, the discrete energy levels are broadened.

Insulator and conductor Insulator atoms Conductor atoms + + + + + + - - - - - - + + + + + + - - - - - - External E-field External E-field + + + + + + - - - - - - + + + + + + - - - - - - Empty energy level - Occupied energy level - Energy level of insulator atoms Energy level of conductor atom

Movement of electrons in a conductor

Charges on a conductor In equilibrium, there is no charge in the interior of a conductor due to repulsive forces between like charges. The charges are bound on the surface of a conductor. The electric field in the interior of a conductor is zero. The electric field emerges on the positive charges and sinks on negative charges. On the surface, tangential component of electric field becomes zero. If non-zero component exist, it induces electric current flow which generates heats on it.

Electric field on a conductor due to external field tangential component normal component +q1 -q1 -q1 Conductor Conductor Tangential component of an external E-field causes a positive charge (+q) to move in the direction of the field. A negative charge (-q) moves in the opposite direction. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

Example : infinite line charge The line charges are on the z-axis and extend to infinity.

Example : infinite surface charge

Example : Volume charge density (z<a) (z>a)

Induction charging Charging a metallic object by induction (that is, the two objects never touch each other). (a) A neutral metallic sphere, with equal numbers of positive and negative charges. (b) The charge on the neutral sphere is redistributed when a charged rubber rod is placed near the sphere. (c) When the sphere is grounded, some of its electrons leave through the ground wire. (d) When the ground connection is removed, the sphere has excess positive charge that is non-uniformly distributed. (e) When the rod is removed, the excess positive charge becomes uniformly distributed over the surface of the sphere.

Induction charging example : inkjet printer Charging a conducting liquid droplet by induction. As the droplet breaks off (d), it retains the charge induced on it by the opposing electrode.