u Electric Field Lines u Eletric Dipoles u Torque on dipole in an external field u Electric Flux u Gauss’ Law

Electric Field Lines u Electric field E(r) is a “vector field”. u How do we visualize a field? »Show the properties of an electric field by drawing the “Electric field lines” u An electric field line pattern indicates both the magnitude and direction of the field. Gravitational Field Point Charge

Electric Field Lines u Direction: »Field lines start on positive charges and end on negative charges. »At any point, draw a tangent to the field line. –This gives the direction of the E-field at that point. –This is also the direction of the force at that point. u Strength: »Given by the number of lines per unit area through a plane perpendicular to the field lines. –Field is the strongest where the lines are close together and weakest where they are far apart –If the lines are uniformly-spaced and parallel, the field is uniform. u Field lines can’t cross!

Conducting Plane

Two Point Charges

Electric Dipoles u Dipole: Two equal and opposite charges separated by length u Example: water molecule. »Charge separation from uneven sharing of electrons by the 2 atoms u Dipole moment: (points from – to + ) u “Polar” molecules have a non-zero dipole moment »Water: (electron shifted  by  4x m, Q=1.6x C) »Water is polar, methane is not u In addition, insulators will generally develop a dipole moment in the presence of an electric field.

Electric Dipoles u Dipole placed in an electric field »Experiences zero net force »Experiences a torque –If the dipole is free to move then the effect of torque is to align the dipole with

Electric Dipoles u Since positive work is done by the field to align the dipole, the potential energy of the dipole decreases u Potential Energy: »Maximum PE when is anti-parallel (opposite) to »Minimum PE when and are in the same direction  Conventional to take potential energy to be zero when u (remember: only changes in potential energy are measurable, so we have freedom to choose where U=0)

Sakurajima Volcano u Streaks of lightning »Is the origin same as lightning accompanying thunderstorms?

Chapter 22: Gauss’ Law

Electric Flux u Flux (origin Latin: “to flow”) »Number of something penetrating a surface. (webster)

Electric Flux u Flux (origin Latin: “to flow”) »Number of something penetrating a surface. (webster) u Electric flux is a measure of the number of electric field lines passing through a surface Direction of Area vector A is normal to the plane of area

Example An arbitrary “closed surface” immersed in an electric field. 90 o

Gauss’ Law u An alternative formulation of Coulomb’s law to compute the E- field due to a distribution of charges. u The integral method of calculating electric field is conceptually simple but can be difficult to implement. u Gauss' Law can be a very powerful and an easy alternative if one can take advantage of the geomteric symmetry of the problem. »example: E-field due to a long line of charge

Karl Friedrich Gauss ( )

Gauss’ Law u Gauss' Law states that the net electric flux through a closed surface is proportional to the charge enclosed by the surface, q enc. u Net flux: »The integral here is over the surface. »The area vector points out from the surface. »The constant  0 is known as the permittivity of free space, and is given by:  0 = 1/(4  k) = 8.85 x C 2 N m 2

Gauss’ Law u The more charge enclosed, the greater the flux through the surface. u The net flux is positive if the net charge enclosed is positive, and negative if the net charge enclosed is negative. u If there is no net charge enclosed by a surface the net flux is zero - any field lines entering the surface must leave the surface somewhere else.

Example: point Charge u Electric Field due to a point charge (Coulomb’s Law) u Field from a point charge “q” is radial »field lines are directed along radii, directly out from or into the charge u Electric Flux: u Independent of radius of the sphere

u Let’s observe the number of field lines passing through the surface in different situations. u We note: »If a surface encloses a net positive charge then more field lines come out of the surface than go into it »If a surface encloses a net negative charge then more field lines go into it than come out »If a surface encloses either no charges or equal numbers of positive and negative charges (ie zero net charge) then the same number of filed lines come out of the surface as go into it. »It appears that the number of field lines through a surface is proportional to the net charge enclosed by the surface (Gauss’ Law) Electric Flux Counting Field Lines through a surface