Physics 2Ba

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Physics 2B(a)

• Electricity and Magnetism

Winter 2004
Lectures by George M. Fuller, UCSD
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George M. Fuller 427 SERF gfuller_at_ucsd.edu 822-1
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Office Hours 200 PM - 330 PM Tuesday 329 SERF
PM
TH 700 PM - 800 PM WLH 2204
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problem assignment TBA
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Electric Potential
Lectures 7, 8, 9
Lectures by George M. Fuller, UCSD
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Conservative forces e.g., gravitation,
electrostatic force
Non-Conservative force e.g., friction force.
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Review of Conservative Forces and Potential Energy
A conservative force is one where the integral of
the inner product (dot product) of the force and
the vector line element vanishes around a closed
loop.
along Path 1 from A to B
along Path 2 from B to A
F
dl
Path 1
B
A
Path 2
along Path 2 from A to B
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The path independence of this integral shows that
it depends only on the endpoints. That, in turn,
suggests that we can associate with it a scalar
quantity, the change in potential energy in going
from A to B
The change in potential energy is the negative of
the work done by the conservative force.
Example Lift a mass m a vertical distance h.
(Uniform gravitational field
with acceleration magnitude g.)
The change in gravitational potential energy is
mgh, independent of the path
taken.
y
B
dl
h
Fmg
A
x
Lectures by George M. Fuller, UCSD
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The relationship between the (vector) Force F and
the potential energy U(x,y,z) at any point is
Partial derivative with respect to, for example,
x treat y and z as constants.
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The electric force is the product of the charge
at some point (x,y,z) and the electric field
vector there
The work done by the electric force in moving a
charge q from point A to point B is
The work done by the electric force per unit
charge in moving from point A to point B is
Lectures by George M. Fuller, UCSD
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The (electrostatic) electric force is a
conservative force, so we can define a potential
energy associated with it. It is most convenient
to work with an electrical potential energy per
unit charge, the so-called electric potential.
(Sometimes we will refer to this as the potential
difference, or sometimes even simply the
potential.) This is analogous to how we defined
the electric field as the force per unit charge.
Since the static electric force is conservative,
the static electric field will have a
vanishing line integral around any closed path
(path independence).
In electrostatic equilibrium
Lectures by George M. Fuller, UCSD
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Path Independence of the Line Integral of a
Static Electric Field
In static conditions (nothing changing in time),
the work done by the electric force per unit
charge around a closed path is
along Path 1 from A to B
along Path 2 from B to A
E
dl
Path 1
B
A
Path 2
along Path 2 from A to B
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Change in electrical potential energy in moving
the charge q from point A to point B
Change in electric potential between points A
and B
Note the change in potential energy in moving
any charge Q from point A to point B is
Lectures by George M. Fuller, UCSD
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What are the dimensions of Electric Potential?
Lectures by George M. Fuller, UCSD
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We can now measure electric field in Volts per
meter.
Lectures by George M. Fuller, UCSD
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One elementary charge e falling through a
potential difference of 1 Volt acquires a kinetic
energy of 1 eV 1 electron volt.
Lectures by George M. Fuller, UCSD
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Does this make physical sense?
Example
Consider moving a positive charge q from point A
to point B along a straight line in a uniform
electric field as shown.
E
A
B
q
q
FqE
dl
The charge was moved uphill.
Lectures by George M. Fuller, UCSD
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The electric potential from a point charge q
The potential difference between two points at
radial distances r1 and r2 , respectively, from
the charge q is
r2
Note the potential difference is, of course,
independent of the path between the
points so we have chosen the most
convenient path to evaluate the
integral.
r1
q
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Really, only potential differences are physically
significant. So, we are free to choose a
convenient zero point for the potential. Call
this the reference point P0 and agree that when
we evaluate the work done by the electric
force we will choose the path through this point.
That is, j(P0 )0.
The potential difference between point P and
point P0 is
Lectures by George M. Fuller, UCSD
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Take the zero of electrical potential for a point
charge to be at infinity. (Reference point at
infinity)
Lectures by George M. Fuller, UCSD
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We can exploit the superposition property of the
electric field even more easily with the
electric potential. Given a distribution of
charge, we can compute the potential at a point
from conveniently chosen increments of charge and
then simply sum these.
Lectures by George M. Fuller, UCSD
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At a particular point in space the
electric potential j is the sum of the
potentials ji from the individual point charges qi
No fuss no muss. No vectors! Simply a straight
algebraic sum of potentials.
Lectures by George M. Fuller, UCSD
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Potential at the origin from point charge qi
z-axis
qi
z
y-axis
x
y
x-axis
Lectures by George M. Fuller, UCSD
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The radius vector from the origin to point (x, y,
z) is
Remember your vector algebra . . .
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Example a very long straight wire with radius
ra carries a line charge
density l. What is the potential difference
between the surface of the
wire and the ground, a distance rb below?
Treat the wire as an infinitely long, charged
rod. Last week we used Gausss Law to show that
the electric field outside such a rod points
radially outward with a magnitude that drops off
inversely like the radial distance from the
rods center line
ra
rb
DVab
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Calculate the potential from a continuous charge
distribution by invoking superposition and
considering the charge distribution as composed
of infinitely many infinitesimal charge elements
dq. These can each be regarded as point charges,
each contributing at point P an increment in
potential dV
Now integrate over the whole charge distribution
to find the potential at point P
Lectures by George M. Fuller, UCSD
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Potential at the origin from a continuous
distribution of charge with charge density r
(x,y,z).
z-axis
z
y-axis
x
y
x-axis
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Example a uniformly charged ring of radius a
and charge Q. find the
electric potential on the axis through the ring.
Assume that the ring is very thin so that we can
regard the charge distribution on it as being a
line charge. Then the line charge density
(charge per unit length)
is l Q/(2p a).
a
x
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By contrast, recall the difficulty of adding the
electric vectors in this example.
Lectures by George M. Fuller, UCSD
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Example a uniformly charged ring of radius a
and charge Q. Find the
electric field on the axis through the ring.
Assume that the ring is very thin so that we can
regard the charge distribution on it as being a
line charge. Then the charge density (charge
per unit length)
is l Q/(2p a).
a
x
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EXAMPLE A ring of radius a 2.6 m carries a
uniformly distributed charge Q 300 nC. A
point charge q -200 nC is located at the center
of the ring. What is the work done by an
external force in moving the point charge out
along the axis through the center of the ring a
distance x 6 m ?
Potential along axis at any position x stemming
from the charges on the ring
a
x
Does the sign make physical sense? Think of
the direction of the electric force. Must oppose
this with external force.
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Example a charged disk of radius a carries a
total charge Q uniformly
distributed over its surface. What is the
potential at a point P on
the disk axis, a distance x from the disk?
surface charge density
Choose a ring-shaped charge element with radius
r and width dr. This has charge increment dq.
This ring gives a potential contribution at
point P
result of previous example
a
r
P
dr
x
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Equipotentials
surfaces of constant potential
The simple physics It takes no work to move a
charge at right angles to
an electric field.
Therefore There can be no potential difference
between two points on a
surface that is everywhere at right angles to the
electric field.
Obviously, such surfaces are equipotential
surfaces.
Example the electric field must meet the surface
of a conductor in electrostatic
equilibrium everywhere at right angles. The
surface of a conductor is an equipotential.
Equipotentials are analogous to contour lines on
a topographic map.
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Draw contours of constant potential (analogous to
contours of altitude on a topographic
map) Where the contour lines are close
together, the potential is changing rapidly with
distance, implying a large electric field.
V -10
V -5
V0
V5
V20
The electric field is normal to contours of
constant potential, with a magnitude inversely
proportional to the spacing of the contour lines.
Lectures by George M. Fuller, UCSD
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The potential will sometimes be denoted by V and
sometimes by j. Specifying the potential at every
point in space is sufficient to determine
the electric field everywhere.
How? Remember that the incremental potential
difference between two points
separated by an infinitesimal (vector)
displacement dl is
Lectures by George M. Fuller, UCSD
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The relationship between the (vector) electric
field E and the potential V(x,y,z) at any point
is
Partial derivative with respect to, for example,
x treat y and z as constants.
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Example calculate the electric field from a
point charge q.
potential at radial distance r from an isolated
point charge q
Lectures by George M. Fuller, UCSD
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Potentials of Charged Conductors
The surface of a conductor in electrostatic
equilibrium is an equipotential.
Example potential at the surface of an isolated
conducting sphere with radius
R and total net charge Q
R
Remember that the electric field vanishes
everywhere inside the conductor in equilibrium.
This implies that there can be no gradient of the
potential inside.
Lectures by George M. Fuller, UCSD
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Example
Consider two charged conducting spheres, one of
radius R1 carrying charge Q1, the other with
radius R2 carrying charge Q2, connected with a
very thin conducting wire. Assume the spheres
are separated by a distance so large that we can
approximate them as isolated. In this limit each
sphere can be regarded as possessing a
uniformly, spherically symmetrically distributed
charge distribution.
E field at conductor surface
can regard this system as one big conductor and,
therefore, an equipotential.
R2
R1
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This example implies that the largest charge
densities and, hence, the largest electric fields
on a conductor will occur where the
surface curvature is biggest (radius of curvature
is smallest).
e.g., a needle or point could be approximated
with our two sphere conductor
Electric fields with strengths above 3 M volts/m
will be strong enough to rip electrons from atoms
in the air, causing the air to become a
conductor. This so-called corona discharge can
sometimes be seen around sharply pointed charged
conductors. The glow comes from recombination of
electrons and ions.
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Example consider three conducting spheres of
radii r1 1 m, r2 3 m,, and r3 6 m all
connected by conducting wires each 400 m long. A
charge 0.1 C is placed on the big sphere. When
electrostatic equilibrium is re-established,
what charge is on the big sphere?
Well we are told that the distances between the
spheres are very large compared to any of their
radii. Therefore, we can approximate the
potential at the surface of any of the spheres as
being from a spherically symmetric
charge distribution (a shell at the surface) with
the radius of that sphere.
Furthermore, we know that in electrostatic
equilibrium all of the (connected) spheres will
have the same potential.
Additionally, we know that charge is conserved.
So in electrostatic equilibrium, when the
charges stop flowing, the sum of the charges on
all the spheres must be the original charge
placed on the big sphere.
So when electrostatic equilibrium obtains again,
the charge on the big sphere is
Lectures by George M. Fuller, UCSD