Transcript PHYS_2326_020509

Examples
A small particle has a charge -5.0 mC and mass 2*10-4 kg. It moves
from point A, where the electric potential is fa =200 V and its speed
is V0=5 m/s, to point B, where electric potential is fb =800 V. What
is the speed at point B? Is it moving faster or slower at B than at A?
E
2
2
A
B
F
mV0
mV
 qa 
 qb
2
2
Vb ~ 7.4 m / s
In Bohr’s model of a hydrogen atom, an electron is considered
moving around a stationary proton in a circle of radius r. Find
electron’s speed; obtain expression for electron’s energy; find total
energy.
11
e2
V2
r

5.3

10
m
U
Fe  ke 2  m
K
r
r
2
T  13.6 eV
T  K U
Calculating Potential from E field
• To calculate potential function from E field
V   
f
i
r r
E  ds

   (E x iˆ  E y ˆj  E z kˆ )  dxiˆ  dyˆj  dzkˆ
f
i


f
i
E x dx  E y dy  E z dz

When calculating potential due to charge distribution, we calculate potential explicitly
if the exact distribution is known.
If we know the electric field as a function of position, we integrate the field.
b

   E d l
a
Generally, in electrostatics it is easier to calculate a potential (scalar) and then find
electric field (vector). In certain situation, Gauss’s law and symmetry consideration
allow for direct field calculations.
Moreover, if applicable, use energy approach
rather than calculating forces directly
(dynamic approach)
Example: Solid conducting sphere
Outside: Potential of the point charge
1
q
V
4 0 r
Inside: E=0, V=const
Potential of Charged Isolated Conductor
• The excess charge on an isolated conductor will distribute itself
so all points of the conductor are the same potential (inside and
surface).
• The surface charge density (and E) is high where the radius of
curvature is small and the surface is convex
• At sharp points or edges  (and thus external E) may reach high
values.
• The potential in a cavity in a conductor is the same as the
potential throughout the conductor and its surface
At the sharp tip (r tends to zero), large
electric field is present even for small
charges.
Lightning rod – has blunt
end to allow larger charge
Corona – glow of air due to gas discharge built-up – higher probability
near the sharp tip. Voltage breakdown of of a lightning strike
the air
Vmax  3  10
6
V /m
Vmax  REmax
Example: Potential between oppositely charged parallel plates
From our previous examples
U ( y )  q0 Ey
V ( y )  Ey
Vab
E
d
Easy way to calculate surface
charge density

 0Vab
d
Remember! Zero potential doesn’t mean the conducting object has no
charge! We can assign zero potential to any place, only difference in
potential makes physical sense
Example: Charged wire
We already know E-field around the wire
only has a radial component
b
1 
rb

ln
Er 
;    E  dr 
2 0 ra
2 0 r
a
Vb = 0 – not a good choice as it follows
Va  
Why so?
We would want to set Vb = 0 at
some distance r0 from the wire
r - some distance from the wire
r0

V
ln
2 0 r
Example: Sphere, uniformly charged inside through volume
r
q  Q 
R
R
3
'
E
Q - volume density of charge

V

r
3 0
R
 ( r  R)
r
 R  r   E dr
2
R
keQ r r
R    3
|R
R 2
keQ
R 
R
Q - total charge
keQ 
r2 
r 
3  2 
2R 
R 
This is given that at infinity  0
Equipotential Surfaces
• Equipotential surface—A surface consisting of a continuous
distribution of points having the same electric potential
• Equipotential surfaces and the E field lines are always
perpendicular to each other
• No work is done moving charges along an equipotential surface
– For a uniform E field the equipotential surfaces are planes
– For a point charge the equipotential surfaces are spheres
Equipotential Surfaces
Potentials at different points are visualized
by equipotential surfaces (just like E-field
lines).
Just like topographic lines (lines of equal
elevations).
E-field lines and equipotential surfaces are
mutually perpendicular
Potential Gradient
b

 a  b   E  d l
On the other hand, we can calculate potential
difference directly
a
a
 a  b   d
b




E  i Ex  j E y  k Ez
Ex  

d  Ex dx  E y dy  Ez dz



: Ey  
: Ez  
x
y
z
E  
Components of E in terms of 
  operator "del"
Frequently, potentials (scalars!) are easier to calculate:
So people would calculate potential and then the field
Superposition for potentials: V = V1 + V2 + …
Example: A positively charged (+q) metal sphere of radius ra is inside
of another metal sphere (-q) of radius rb. Find potential at different points
inside and outside of the sphere.
a) r  ra : b)ra  r  rb : c)r  rb
1
-q
a)
2
V2 (r ) 
V1 (r ) 
+q
q
4 0rb
q
4 0ra
Total V=V1+V2
b)
Electric field between spheres
V (r ) 
q 1 1
  
4 0  r rb 

E
r
V (r ) 
c)
q 1 1
  
4 0  ra rb 
V 0
Method of images
How to calculate E-field in the vicinity
of conducting surface given that there
are induced charges on that surface?
h

 0
q 1 1 
  '
4 0  r r 
Everywhere on the conductive surface
Grounded conductor φ=const
Uniqueness theorem in electrostatics : if we know space charge distribution and
boundary conditions such as potentials at all conductors, the electric field is uniquely
defined in space
We can remove the surface and replace it with single point charge –q at
a distance h behind the surface.
Attractive force between charges
q2
F
4 0 (2h) 2
Definitions
• Voltage—potential difference between two points in space (or a
circuit)
• Capacitor—device to store energy as potential energy in an E
field
• Capacitance—the charge on the plates of a capacitor divided by
the potential difference of the plates C = q/V
• Farad—unit of capacitance, 1F = 1 C/V. This is a very large unit
of capacitance, in practice we use mF (10-6) or pF (10-12)
Definitions cont
• Electric circuit—a path through which charge can flow
• Battery—device maintaining a potential difference V
between its terminals by means of an internal
electrochemical reaction.
• Terminals—points at which charge can enter or leave a
battery
Capacitors
• A capacitor consists of two conductors called plates which get equal
but opposite charges on them
• The capacitance of a capacitor C = q/V is a constant of
proportionality between q and V and is totally independent of q and
V
• The capacitance just depends on the geometry of the capacitor, not q
and V
• To charge a capacitor, it is placed in an electric circuit with a source
of potential difference or a battery
CAPACITANCE AND CAPACITORS
Capacitor: two conductors separated by
insulator and charged by opposite and
equal charges (one of the conductors can be
at infinity)
Used to store charge and electrostatic
energy
Superposition / Linearity: Fields, potentials and potential
differences, or voltages (V), are proportional to charge
magnitudes (Q)
C
Q
V
(all taken positive, V-voltage between plates)
Capacitance C (1 Farad = 1 Coulomb / 1 Volt) is
determined by pure geometry (and insulator properties)
1 Farad IS very BIG: Earth’s C < 1 mF
Parallel plate capacitor
Energy stored in a capacitor is related to the E-field between the plates
Electric energy can be regarded as stored in the field itself.
This further suggests that E-field is the separate entity that may exist alongside
charges.
density   charge Q / area S
E

Q

;
 0  0S
C
V  Ed 
 0S
Qd
 0S
Generally, we find the potential difference
Vab between conductors for a certain
charge Q
Point charge potential difference ~ Q
d
This is generally true for all capacitances
Capacitance configurations
Cylindrical capacitor
Spherical Capacitance
b
dr
1 1
V  keQ  2  keQ(  )
r
a b
a
b
dr
Q b
V  2ke    2ke ln( )
r
l
a
a
C
ab
C
ke (b  a)
l
With b  , C  a / ke -
b
2ke ln( )
a
capacitance of an individual sphere
Definitions
• Equivalent Capacitor—a single capacitor that has the same
capacitance as a combination of capacitors.
• Parallel Circuit—a circuit in which a potential difference applied
across a combination of circuit elements results in the potential
difference being applied across each element.
• Series Circuit—a circuit in which a potential difference applied
across a combination of circuit elements is the sum of the
resulting potential differences across each element.
Capacitors in Series
Q
Q
Vac  V1  ; Vcb  V2 
C1
C2
Total voltage V  V1  V2
Equivalent
1 V
1
1
 

C Q C1 C2
Capacitors in Parallel
Total charge
Q  Q1  Q2
Equivalent
C
Q
 C1  C2
V
Example: Voltage before and after
Initially capacitors are charged by
the same voltage but of opposite polarity :
Q1i  C1Vi ; Q2i  C2Vi
Total charge Q  Q1i  Q2i  Q1 f  Q2 f
Equivalent
C  C1  C2
Q C1  C2
Voltage after : V f  
Vi
C C1  C2
Applications
In 1995, the microprocessor unit on the microwave imager on the DMSP
F13 spacecraft locked up - occurred ~5 s after spacecraft began to charge
up in the auroral zone in an auroral arc. Attributed to high-level charging of
spacecraft surface and subsequent discharge.
Spacecraft surfaces are generally covered with thermal blankets - outer
layer some dielectric material - typically Kapton or Teflon. Deposition of
charge on surface of spacecraft known as surface charging. Incident
electrons below about 100 keV penetrate the material to a depth of a few
microns, where they form a space charge layer - builds up until breakdown
occurs accompanied by material vaporization and ionization. A discharge is
initiated - propagates across surface or through the material, removing part
of bound charge. Typically occur in holes, seams, cracks, or edges - have
been know to seriously damage spacecraft components.
Thermal blankets composed of layers of
dieletric material with vapor
deposited aluminum (VDA)
between each layer. On
DMSP, VDA between layers
(22) not grounded - serve as
plates of a set of 22 parallelplate capacitors - top plate
consists of electrons buried
in top few microns of Teflon.
1 22 1

C i1 c i
C/A=7.310-9 F/m2
 of a parallel plate capacitor to some voltage with
Time to charge outer surface
respect to spacecraft frame is:
t
CV
i(1   )A
Laboratory measurements for Teflon:
-discharge at 3 kV in a 20 keV electron beam
-secondary electron yield () at 20 keV ~ 0.2
Given measured incident precipitating current density of i = 4.8 mA/m2, the time
to reach breakdown voltage for conditions experienced by DMSP F13 is 5.7 s this is the time after the spacecraft began to charge up that the lockup
occurred.
If the VDA layer on the bottom side of the outer Teflon layer were grounded to
the spacecraft frame, the capacitance would have been 3.510-7 F/m2 and the
charging time would have been 132 s - no discharge would have occurred.