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Unformatted text preview: Physics 142 7/18/2011 The Electric Flux
The electric flux measures the amount of electric field
passing through a surface of area A whose normal to the
surface is tilted at angle θ from the field. We can define the electric flux more concisely using the
dotproduct: The total electric flux through this box is 0%
0%
0%
0%
0% Nm2/C.
B. 4 Nm2/C.
C. 2 Nm2/C
D. 1 Nm2/C.
E. 0 Nm2/C.
A. 6 Physics 142 Summer 2011 Dr. Nick Cummings 1 Physics 142 7/18/2011 Electric Flux
rr
so
Φ e = E ⋅ A so units are N2
⋅m
C Area
Area vector could point either way
– Sign ambiguous so far If
If the electric field lies in the plane of the
surface flux is zero
If Efield is perpendicular flux is just EA
If E Physics 142 Summer 2011 Electric Flux and Field Lines
Geometrically: electric flux proportional to #
of field lines passing through surface
– Density of field lines proportional to field
strength
– (density of line)* area = # of lines Physics 142 Summer 2011 Dr. Nick Cummings 2 Physics 142 7/18/2011 Computing Flux
r r Computing
Computing flux is generally hard if E or A
are changing at the surface
– Efield changing direction
– Surface curved BUT….
BUT…. Easy if everywhere on the surface
the field is either perpendicular or parallel Physics 142 Summer 2011 Flux for a Closed Surface
For
For a closed surface define area vector to
point out
– Total flux is (flux out) – (flux in) Geometrically:
Geometrically: (# of field lines out)(# of
out)field lines in)
– Sources outside don’t contribute to flux
– It doesn’t matter where the source is
– Size and shape of surface don’t matter
Physics 142 Summer 2011 Example: Point Source Flux
Consider a point charge Q and a surface that
is a sphere of radius R centered on that
charge. What is the flux through that
sphere? Physics 142 Summer 2011 Dr. Nick Cummings 3 Physics 142 7/18/2011 Gauss’s Law
For any closed surface enclosing total charge Qin,the net
electric flux through the surface is This result for the electric flux is known as Gauss’s Law. This box contains 0%A.
0%B.
0%C.
0%D.
D.
0%E.
E. a net positive charge.
a net negative charge.
a negative charge.
a positive charge.
no net charge. Physics 142 Summer 2011 These
These are twodimensional cross sections through threetwothreedimensional closed spheres and a cube. Rank order, from largest
to smallest, the electric fluxes Φa to Φe through surfaces a to e. 0% A. 0% B. 0% C. 0% D. 0% E. Φa > Φc > Φb > Φd > Φe
Φb = Φe > Φa = Φc = Φd
Φe > Φd > Φb > Φc > Φa
Φb > Φa > Φc > Φe > Φd
Φd = Φe > Φc > Φa = Φb
Physics 142 Summer 2011 Dr. Nick Cummings 4 Physics 142 7/18/2011 Who Cares About Flux?
Gauss’
Gauss’ law relates charge and flux, but who
cares about flux?
– Generally flux integral is hard to compute
– Often we want to know the Efield value at a pt
E If
If the source is symmetrical, there may be a
surface for which the Efield is always
Eperpendicular or parallel
– Calculating flux is easy
– Can use flux to solve for Efield value
EPhysics 142 Summer 2011 Using Flux to Find the Field
Gauss’
Gauss’ law applies whether the surface is a
physical object or just imaginary
With
With enough symmetry we can construct an
imaginary surface where
– Easy to calculate flux
– Possible to solve for Efield value
E The
The imaginary surface is called a “Gaussian
surface”
Physics 142 Summer 2011 Example: Charged Sphere
Consider a sphere with charge Q distributed
uniformly throughout it. Using Gauss’ law
and symmetry, derive the formula for the
electric field. Physics 142 Summer 2011 Dr. Nick Cummings 5 Physics 142 7/18/2011 Example: Charged Cylinder
Consider an infinite cylinder with uniform
charge per unit length λ distributed
uniformly throughout it. Using Gauss’ law
and symmetry, derive the formula for the
electric field. Physics 142 Summer 2011 Which Gaussian surface would allow you to use
Gauss’s law to determine the electric field outside
a uniformly charged cube?
A cube whose center coincides with the center of the
charged cube and which has parallel faces.
B. A sphere whose center coincides with the center of
the charged cube.
0%
C. Neither A nor B.
D. Either A or B.
A. A. B. C. D. Physics 142 Summer 2011 Dr. Nick Cummings 6 ...
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