Download Lecture 6 - UConn Physics

Physics 1502: Lecture 6
Today’s Agenda
• Announcements:
– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, solutions etc.
• Homework #2:
– On Masterphysics today: due Friday
– Go to masteringphysics.com
• Labs: Begin this week
Today’s Topic :
• End of Chapter 22: Electric potential
– Equipotentials and Conductors
– Electric Potential Energy
» of Charge in External Electric Field
• Chapter 23: Electrostatic energy
– Definition and concept
Electric Potential
V
Q
4pe0 r
Q
4pe0 R
R
r
R
C
R
B
r
B
q
r
A
A
path independence
equipotentials
Electric Potential
• We define the electric potential difference as:
• independent of q0
• independent of path
• We define the electric potential of a point in space as the
potential difference between that point and a reference point.
• a good reference point is infinity ... we typically set V = 0
• the electric potential is then defined as:
The Bottom Line
If we know the electric field E everywhere,
allows us to calculate the potential function V everywhere
(define VA = 0 above)
If we know the potential function V everywhere,
1
allows us to calculate the electric field E everywhere.
Units for Potential! 1 Joule/Coulomb = 1 VOLT
z
Lecture 6, ACT 1
• Consider the dipole shown at the right.
– Fix r = r0 >> a
– Define qmax such that the polar
component of the electric field has its
maximum value (for r=r0).
1
What is qmax?
(a) qmax = 0
(b) qmax = 45
+q
r1
r
aq
a
-q
(c) qmax = 90
r2
Dipole Field
y=
Etot
z
+q
a q
r
E
q
0
a
Er
-q
0
p/
p

p
p/
x=
Equipotentials
Defined as: The locus of points with the same potential.
•
Example: for a point charge, the equipotentials are
spheres centered on the charge.
• GENERAL PROPERTY:
– The Electric Field is always perpendicular to an
Equipotential Surface.
• Why??
The gradient (  ) says E is in the direction of
max rate of change.
Along the surface, there is NO change in V
(it’s an equipotential!)
So, there is NO E component along the surface
either… E must therefore be normal to surface
Dipole
Equipotentials
Conductors
+
+
+
+
+
+
+
+
+
•
Claim
+
+
+
+
+
The surface of a conductor is always an equipotential surface
(in fact, the entire conductor is an equipotential)
•
Why??
If surface were not equipotential, there would be an Electric Field
component parallel to the surface and the charges would move!!
•
Note
Positive charges move from regions of higher potential to lower
potential (move from high potential energy to lower PE).
Equilibrium means charges rearrange so potentials equal.
Charge on Conductors?
• How is charge distributed on the surface of a
conductor?
– KEY: Must produce E=0 inside the conductor and E normal to the
surface .
Spherical example (with little off-center charge):
+ + +
+
- -- +
- +
+ -+q - +
+ - +
+ - +
+
+ + +
E=0 inside conducting shell.
charge density induced on
inner surface non-uniform.
charge density induced on
outer surface uniform
E outside has spherical
symmetry centered on spherical
conducting shell.
A Point Charge Near
Conducting Plane
q
+
a
V=0
- - - - - - -- - - - - - -- --- - - - - - - - - - - - - - - - - - -
A Point Charge Near
Conducting Plane
q
+
a
The magnitude of the force is
q2
F
4pe0 2a 2
1
Image Charge
The test charge is attracted to a conducting plane
Charge on Conductor
• How is the charge distributed on a non-spherical
conductor?? Claim largest charge density at smallest
radius of curvature.
• 2 spheres, connected by a wire, “far” apart
• Both at same potential
rL
rS
But:

Smaller sphere
has the larger
surface charge
density !
Equipotential Example
•
Field lines more closely
spaced near end with most
curvature .
•
Field lines ^ to surface near
the surface (since surface is
equipotential).
•
Equipotentials have similar
shape as surface near the
surface.
•
Equipotentials will look more
circular (spherical) at large r.
Electric Potential Energy
• The Coulomb force is a CONSERVATIVE force (i.e. the work
done by it on a particle which moves around a closed path
returning to its initial position is ZERO.)
• Therefore, a particle moving under the influence of the
Coulomb force is said to have an electric potential energy
defined by:
this “q” is the ‘test charge”
in other examples...
• The total energy (kinetic + electric potential) is then conserved
for a charged particle moving under the influence of the
Coulomb force.
St Elmo’s Fire
Van de Graaff Generator
Schematic view of a classical Van de
Graaff generator.
1) hollow metal sphere
2) upper electrode
3) upper roller (for example in acrylic
glass)
4) side of the belt with positive charges
5) opposite side of the belt with negative
charges
6) lower roller (metal)
7) lower electrode (ground)
8) spherical device with negative charges,
used to discharge the main sphere
9) spark produced by the difference of pot
Lightning
Energy Units
MKS:
U = QV 
for particles (e, p, ...)
1 coulomb-volt = 1 joule
1 eV
= 1.6x10-19 joules
Accelerators
• Electrostatic: VandeGraaff
electrons  100 keV ( 105 eV)
• Electromagnetic: Fermilab
protons  1TeV ( 1012 eV)
Definitions & Examples
A
++++
d -----
a
b
L
a
C3
b
C1
C2

a
b
C
Overview
• Definition of Capacitance
• Example Calculations
(1) Parallel Plate Capacitor
(2) Cylindrical Capacitor
(3) Isolated Sphere
• Energy stored in capacitors
• Dielectrics
• Capacitors in Circuits
Text Reference: Chapter 23
Capacitance
• A capacitor is a device whose purpose is to store electrical
energy which can then be released in a controlled manner
during a short period of time.
+
-
• A capacitor consists of 2 spatially separated conductors
which can be charged to +Q and -Q respectively.
• The capacitance is defined as the ratio of the charge on one
conductor of the capacitor to the potential difference
between the conductors.
• Is this a "good" definition? Does the capacitance
belong only to the capacitor, independent of the
charge and voltage ?
Example 1:
Parallel Plate Capacitor
• Calculate the capacitance. We
assume +s, - s charge densities
on each plate with potential
difference V:
• Need Q:
• Need V:
from defn:
– Use Gauss’ Law to find E
A
++++
d -----
Recall:
Two Infinite Sheets
(into screen)
• Field outside the sheets is zero
• Gaussian surface encloses
zero net charge
• Field inside sheets is not zero:
• Gaussian surface encloses
non-zero net charge
+
s
+
E=0 +
+
+
A
+
+
+
+
A+
+
+
s
- E=0
E
Example 1:
Parallel Plate Capacitor
• Calculate the capacitance:
Assume +Q,-Q on plates with
potential difference V.
A
++++
d -----

• As hoped for, the capacitance of this capacitor
depends only on its geometry (A,d).
Dimensions of capacitance
• C = Q/V => [C] = F(arad) = C/V = [Q/V]
• Example: Two plates, A = 10cm x 10cm
d = 1cm apart
=> C = Ae0/d =
= 0.01m2/0.01m * 8.852e-12 C2/Jm
= 8.852e-12 F
Lecture 6 - ACT 2
• Suppose the capacitor shown here is
charged to Q and then the battery
disconnected.
• Now suppose I pull the plates
further apart so that the final
separation is d1. d1 > d
A
++++
d -----
A
++++
d1
-----
If the initial capacitance is C0 and the final capacitance is C1, is …
A) C1 > C0
B) C1 = C0
C) C1 < C0