Electric Field Lines

Eletric Dipoles Torque on dipole in an external field

Electric Flux Gauss’ Law

Electric Field Lines Electric field E(r) is a “vector field”. How do we visualize a field?

» Show the properties of an electric field by drawing the “Electric field lines”

An electric field line pattern indicates both the magnitude and direction of the field.

Gravitational Field

Point Charge

Electric Field Lines 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.

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.

Field lines can’t cross!

Conducting Plane

Two Point Charges

Electric Dipoles

Dipole: Two equal and opposite charges separated by length

Example: water molecule.» Charge separation from uneven

sharing of electrons by the 2 atoms

Dipole moment: (points from – to + )

“Polar” molecules have a non-zero dipole moment» Water: (electron shifted by 4x10-11m, Q=1.6x10-19 C)

» Water is polar, methane is not

In addition, insulators will generally develop a dipole moment in the presence of an electric field.

Electric Dipoles

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

Since positive work is done by the field to align the dipole, the potential energy of the dipole decreases

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 (remember: only changes in potential energy are measurable, so

we have freedom to choose where U=0)

Sakurajima Volcano

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

Chapter 22: Gauss’ Law

Electric Flux Flux (origin Latin: “to flow”)

» Number of something penetrating a surface. (webster)

Electric Flux Flux (origin Latin: “to flow”)

» Number of something penetrating a surface. (webster)

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.

90o

Gauss’ Law

An alternative formulation of Coulomb’s law to compute the E-field due to a distribution of charges.

The integral method of calculating electric field is conceptually simple but can be difficult to implement.

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 (1777-1855)

Gauss’ Law Gauss' Law states that the net electric flux through a

closed surface is proportional to the charge enclosed by the surface, qenc.

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 10-12 C2 N m2

Gauss’ Law The more charge enclosed, the greater the flux through

the surface. The net flux is positive if the net charge enclosed is

positive, and negative if the net charge enclosed is negative.

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 Electric Field due to a point charge (Coulomb’s Law)

Field from a point charge “q” is radial » field lines are directed along radii, directly out from or into the charge

Electric Flux:

Independent of radius of the sphere

Let’s observe the number of field lines passing through the surface in different situations.

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 FluxCounting Field Lines through a surface