Physics 1202: Lecture 4Today’s Agenda
• Announcements:– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, solutions etc.
• Homework #1:Homework #1:– On Masterphysics: due this coming FridayOn Masterphysics: due this coming Friday
– Go to masteringphysics.com and register– Course ID: MPCOTE62465
• Labs: Begin next week
Today’s Topic :
• Chapter 16: Electric energy & potential– Definition– How to compute them– Point charges– Capacitors
Electric potential EnergyRecall 1201
•Total mechanical energy– Constant for conservative forces
•Potential energy U– Depends only on position (ex: U = mgy)– Change in U is independent of path
kinetic potential
U2 , y2
U1 , y1
Electric potential Energy•Gravitational force (magnitude)
•Gravitational Potential energy U
•By analogy:
Electric force Electric potential energy
Electric potential Energy•Meaning: recall
•Total energy is conserved– Variation of U with r variation of kinetic energy
•For multiple charges– Simple sum – Ex: 3 charges
q1
q3
q2
r13
r12
r23
Energy UnitsMKS: U = QV 1 coulomb-volt = 1 joule
for particles (e, p, ...) 1 eV = 1.6x10-19 joules
Accelerators• Electrostatic: VandeGraaff
electrons 100 keV ( 105 eV)
• Electromagnetic: Fermilab
protons 1TeV ( 1012 eV)
Electric Potential• By analogy with the electric field
• Defined using a test charge q0
• We define a potential V due to a charge q
– Using potential energy by charge q and a test charge q0
Electric Potential• Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field E.
• Since there will be a force on the charge due to E, a certain amount of work WAB will have to be done to accomplish this task. We define the electric potential difference as:
• The potential difference is meaningful
• Because only potential energy difference is meaningful
• U=V=0 can be chosen arbitrarily (like for gravity)
A B
q0E
Lecture 4, ACT 1• A single charge ( Q = -1C) is fixed at the origin.
Define point A at x = + 5m and point B at x = +2m.
– What is the sign of the potential difference between A and B? (VAB VB - VA )
(a) VAB < (b) VAB = (c) VAB >
x-1C AB
• The simplest way to get the sign of the potential difference is to imagine placing a positive charge at point A and determining whether positive or negative work would be done in moving the charge to point B.
• A positive charge at A would be attracted to the -1C charge; therefore NEGATIVE work would be done to move the charge from A to B.
E
Electric Potential• 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:
• for a point charge, the formula is:
Potential from N charges
The potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately.
xr1
r2 r3
q1
q3
q2
Electric Dipole z
a
a
+q
-q
r
r1r2
The potential is much easier to calculate than the field since it is an algebraic sum of 2 scalar terms.
• Rewrite this for special case r>>a:
Can we use this potential somehow to calculate the E field of a dipole?
(remember how messy the direct calculation was?)
r2-r1
E from V?• We can obtain the electric field E from the potential V by
inverting our previous relation between E and V:
• Consider 2 plates and a charge q
• force on q
• Work done on q
++
++
++
++
++
++
++
++
++
++
++
++
++
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
+
F
• But work-energy theorem
• Conservative force
E from V?• We can obtain the electric field E from the potential V by
inverting our previous relation between E and V:
• We have
++
++
++
++
++
++
++
++
++
++
++
++
++
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
- -
+
F
• So that
• Therefore
Equipotentials
• GENERAL PROPERTY: – The Electric Field is always perpendicular to an
Equipotential Surface.
• Why??
Dipole Equipotentials
Defined as: The locus of points with the same potential.
• Example: for a point charge, the equipotentials are spheres centered on the charge.
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
• ClaimThe 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!!
• NotePositive 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.
Conductors
+ +
+ +
+ +
+ + +
+ + +
+ +
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):
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.
+q
A Point Charge Near Conducting Plane
+
a
q
- - -- -- - - -- - - - -- --- ---------------- --- -- ---- --------V=0
A Point Charge Near Conducting Plane
+
-
a
q
The magnitude of the force is
The test charge is attracted to a conducting plane
Image Charge