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Electric Field Lines
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Eletric Dipoles
Torque on dipole in an external field
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Electric Flux
Gauss’ Law
Electric Field Lines
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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”
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An electric field line pattern indicates both the magnitude and
direction of the field.
Point Charge
Gravitational Field
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
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Dipole: Two equal and opposite charges separated by length
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Example: water molecule.
» Charge separation from uneven
sharing of electrons by the 2 atoms
(points from – to + )
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Dipole moment:
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“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
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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:
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» 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
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Streaks of lightning
» Is the origin same as lightning accompanying thunderstorms?
Chapter 22: Gauss’ Law
Electric Flux
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Flux (origin Latin: “to flow”)
» Number of something penetrating a surface. (webster)
Electric Flux
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Flux (origin Latin: “to flow”)
» Number of something penetrating a surface. (webster)
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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
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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
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Gauss' Law states that the net electric flux through a
closed surface is proportional to the charge enclosed
by the surface, qenc.
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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.
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Example: point Charge
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Electric Field due to a point charge (Coulomb’s Law)
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Field from a point charge “q” is radial
» field lines are directed along radii, directly out from or into the charge
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Electric Flux:
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Independent of radius of the sphere
Electric Flux
Counting Field Lines through a surface
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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)