Download The electric field of an infinite plane of charge

PHYS117B: Lecture 6
Electric field in planar geometry.
Electric potential energy.
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Last lecture:
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1/22/2007
Properties of conductors and insulators in electrostatic
equilibrium
 E = 0 inside the conductor and all excess charges are
on the surface
Used Gauss’s law to find the field in and out of spheres
(conductors and insulators) … and similarly we can do
spherical shells and spheres inside spherical shells
 The electric field outside a sphere is = to the E of a point
charge located in the center
We “played’ with cylinders the previous time
J.Velkovska, PHYS117B
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The electric field of an infinite charged plane
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Use symmetry:
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The field is ┴ to the
surface
Direction: away from
positive charge, and
toward a negative charge
Use Gauss’s law to
determine the
magnitude of the field
J.Velkovska, PHYS117B
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Here’s how we do it: … as easy as 1,2,3
Choose a Gaussian surface:
1.
a cylinder would work: the field
is ┴ to the area vector on the
sides and ║ to the area vector
on the top and the bottom of the
cylinder
a cube or a parallelogram with
sides ║to the surface would
work, too
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Evaluate the flux through the
surface and the enclosed
charge
2.
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EA +EA = 2EA
Qencl = σ A
Apply Gauss’s law:
3.
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E = σ/ 2ε0
The electric field of an infinite plane of charge does NOT depend on
the distance from the plane, but ONLY on the surface charge density
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J.Velkovska, PHYS117B
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Now add a second plane with opposite
charge: parallel plate capacitor
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For the negatively charged plane:
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Flux : - 2EA
Charge: -σ A
E = σ/ 2ε0 , pointing towards the plane
Use superposition to find the field between the plates and outside
the plates:
E=0 , outside the plates
 E = σ/ ε0
 Direction : from + to 1/22/2007
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J.Velkovska, PHYS117B
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Use the properties of conductors and
Gauss’s law: expel the field from some
region in space
When a lightening strikes
You are safe inside your car
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J.Velkovska, PHYS117B
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Electric field shielding has multiple uses
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If you want to measure the gravitational force
between 2 objects (Cavendish balance), you need
to make sure that electric forces don’t distort your
measurement
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Put the one of the objects in a light weight metal mesh
(Faraday cage) to screen any stray electric fields
Use a coaxial cable ( has a central conductor
surrounded by a metal braid which is connected to
ground) to transmit sensitive electric signals
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OK, we know how to get the Electric field
in almost any configuration, but what does
this tell us about how objects in nature
interact ?
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Well, we know the definition:
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Electric field = Force/unit charge
So if we know E, we can find the force on a charge that is
placed inside the field
We can use F= ma and kinematics to find how this charge
will move inside the field ( we did this for homework)
Today: we will use conservation of energy – a very
powerful approach !
1/22/2007
J.Velkovska, PHYS117B
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Electric potential energy
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The potential energy is a measure of the interactions in the
system
Define: the change in potential energy by the WORK done
by the forces of interaction as the system moves from one
configuration to another
Electric force is a conservative force: the work doesn’t
depend on the path taken, but only on the initial and final
configuration => Conservation of energy
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How can the path not matter ?
Charge q2 moves in the field of q1
Well, the work is not just
Force multiplied by
displacement,
it is the SCALAR Product
between the two.
f
W   F  dl
i
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Electric potential energy in a uniform
field: a charge inside a parallel plate
capacitor
b
W   F  dl
U  qEy
a
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The potential energy of two point charges
f
W   F  dl  kq0 q i dr / r 2
f
i
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J.Velkovska, PHYS117B
The force is along the radius
The work ( and the change in the
potential energy) depends only on
the initial and final configuration
The potential energy depends on
the distance between the charges
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If we have a collection of charges:
f
W   F  dl
i
F  F  F  F  .....
1
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2
3
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Graph the potential energy of two point charges
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U depends on 1/r and on the relative sign of the charges
Defined up to a constant. We take U = 0 when the charges
are infinitely far apart. Think of it as “no interaction”.
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Conservation of Energy in 2 charge system
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Total mechanical energy Emech = const
Emech > 0 , the particles can escape each other
Emech < 0, bound system
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2 examples ( done on the blackboard)
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Distance of closest approach for 2 like
charges
Escape velocity for 2 unlike charges
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