Transcript Lecture_5

Chapter 23
Electric Potential
Basics
V  a  V b   E  d
a
b
parallels definition for ANY conservative force

 dV   E  d   Ex dx  E y dy  Ez dz

V
V
V
 Ex  
, Ey  
, Ez  
x
y
z
Choice of where V  r0   0 is arbitrary
Typically, if charge distribution is finite a good chioce is r0  
For a point charge Q:
  r '  d 3r '
Q
1 Q
V r  k 
 V  r    k
or

r 4 0 r
r r'
Qi
i k r  r
i
23-6 Electric Dipole Potential
The potential due to an
electric dipole is just the sum
of the potentials due to each
charge, and can be calculated
exactly. For distances large
compared to the charge
separation:
23-7 E Determined from V
If we know the field, we can determine the
potential by integrating. Inverting this
process, if we know the potential, we can
find the field by differentiating:
This is a vector differential equation;
here it is in component form:
23-6 Electric Dipole Potential
The potential due to an
electric dipole is just the sum
of the potentials due to each
charge, and can be calculated
exactly. For distances large
compared to the charge
separation:
yˆ

Can always choose system
such that P lies in x-y plane.
 xˆ
23-6 Electric Dipole Potential -- Field

 1 
x
x
2
2 3 2
V  x, y   kp  2
 kpx  x  y 

  kp
32
2 
2
2
2
2


 x  y  x  y 
x  y 
 2

V
 3 2
2 3 2
2 5 2

 kp  x  y   x     x  y 
2x 
x
 2


kp
kp
2
2
2
2
2





x

y

3
x

y

2
x

2
2 52 
2
2 52 
x  y 
x  y 
 
 Ex  
V
kp
2
2



2
x

y
52 
2
2
x  x  y 
V
xy
 3 2
2 5 2
and
 kpx     x  y 
2 y  3kp
2
2 52
y
2


x  y 
 
 Ey  
V
xy
 3kp
2
2 52
y
x  y 
Summary of Chapter 23
• Electric potential is potential energy per
unit charge:
• Potential difference between two points:
• Potential of a point charge:
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Summary of Chapter 23
• Equipotential: line or surface along which
potential is the same.
• Electric dipole potential is proportional to 1/r2.
• To find the field from the potential:
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Chapter 24
Capacitance, Dielectrics,
Electric Energy Storage
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Units of Chapter 24
• Capacitors
• Determination of Capacitance
• Capacitors in Series and Parallel
• Electric Energy Storage
• Dielectrics
• Molecular Description of Dielectrics
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24-1 Capacitors
A capacitor consists of two conductors
that are close but not touching. A
capacitor has the ability to store electric
charge.
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24-1 Capacitors
Parallel-plate capacitor connected to battery. (b)
is a circuit diagram.
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24-1 Capacitors
When a capacitor is connected to a battery, the
charge on its plates is proportional to the
voltage:
The quantity C is called the capacitance.
Unit of capacitance: the farad (F):
1 F = 1 C/V.
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24-2 Determination of Capacitance
For a parallel-plate capacitor
as shown, the field between
the plates is
E = Q/ε0A.
Integrating along a path
between the plates gives the
potential difference:
Vba = Qd/ε0A.
This gives the capacitance:
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24-2 Determination of Capacitance
Example 24-1: Capacitor calculations.
(a) Calculate the capacitance of a parallel-plate
capacitor whose plates are 20 cm × 3.0 cm and
are separated by a 1.0-mm air gap. (b) What is
the charge on each plate if a 12-V battery is
connected across the two plates? (c) What is
the electric field between the plates? (d)
Estimate the area of the plates needed to
achieve a capacitance of 1 F, given the same
air gap d.
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24-2 Determination of Capacitance
Capacitors are now made with capacitances
of 1 farad or more, but they are not parallelplate capacitors. Instead, they are activated
carbon, which acts as a capacitor on a very
small scale. The capacitance of 0.1 g of
activated carbon is about 1 farad.
Some computer keyboards
use capacitors;
depressing the
key changes the
capacitance, which
is detected in a
circuit.
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24-2 Determination of Capacitance
Example 24-2: Cylindrical
capacitor.
A cylindrical capacitor consists of a
cylinder (or wire) of radius Rb
surrounded by a coaxial cylindrical
shell of inner radius Ra. Both
cylinders have length l which we
assume is much greater than the
separation of the cylinders, so we can
neglect end effects. The capacitor is
charged (by connecting it to a battery)
so that one cylinder has a charge +Q
(say, the inner one) and the other one
a charge –Q. Determine a formula for
the capacitance.
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24-2 Determination of Capacitance
Example 24-3: Spherical capacitor.
A spherical capacitor
consists of two thin
concentric spherical
conducting shells of radius
ra and rb as shown. The
inner shell carries a
uniformly distributed
charge Q on its surface,
and the outer shell an
equal but opposite charge –Q.
Determine the capacitance of the
two shells.
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24-2 Determination of Capacitance
Consider two small (radius r) spheres a
large distance (R) apart. What is their
capacitance?
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24-2 Determination of Capacitance
Consider two small (radius r) spheres a
large distance (R) apart. What is their
capacitance?
V  
Rr
 
Rr
r
r
E  dx   
Rr
r
E Q  x  dx  1
 k  Q 
kQ  1 
dx 

2 dx 
2
x
 R  x  

Rr
Rr
 R  r dx
Rr
dx 
  1 
 1  
 kQ  


2   kQ  
2

 
r
r
x
x
x

R



R

x

 
r
r



 1
1
1
1  2kQ
 kQ 
 


r

R
r
R

r

R
r

R
r


C 
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Q r

 2 0 r
V 2k
Questions?
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