Chapter 17 Electric Potential and Electric Energy; Capacitance.
ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the...
-
Upload
claribel-hodges -
Category
Documents
-
view
216 -
download
0
Transcript of ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the...
![Page 1: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/1.jpg)
ELECTRIC POTENTIALELECTRIC POTENTIAL
Spring , 2008Spring , 2008
![Page 2: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/2.jpg)
Chapter 24 Electric PotentialIn this chapter we will define the electric potential ( symbol V ) associated with the electric force and accomplish the following tasks:
Calculate V if we know the corresponding electric field Calculate the electric field if we know the corresponding potential V Determine the potential V generated by a point charge Determine the potential V generated by a continuous charge distribution Determine the electric potential energy U of a system of charges Define the notion of an equipotential surface Explore the geometric relationship between equipotential surfaces and electric field lines Explore the potential of a charged isolated conductor
(24 - 1)
![Page 3: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/3.jpg)
Picture a Region ofPicture a Region ofspace Where there is an space Where there is an
Electric FieldElectric Field
Imagine there is a particle of charge Imagine there is a particle of charge q at some location.q at some location.
Imagine that the particle must be Imagine that the particle must be moved to another spot within the moved to another spot within the field.field.
WorkWork must be done in order to must be done in order to accomplish this.accomplish this.
![Page 4: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/4.jpg)
What (or who) must do this What (or who) must do this work?work?
An external agent An external agent (person)(person)
The Field itselfThe Field itself Either of the aboveEither of the above
![Page 5: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/5.jpg)
Electric Potential
We will be dealing with Work Energy & Conservation
Work must be done to move a charge in an electric field. Let’s do a demo ….
![Page 6: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/6.jpg)
I need some help.
![Page 7: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/7.jpg)
What we will do ….
For the moment, assume the charge has MASS. (It may not.) Assume the charge is initially stationary. The charge is to be moved to the left. The charge is to be moved at CONSTANT velocity.
+charge
E
Mr. ExternalMrs. Fields
![Page 8: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/8.jpg)
During this process, who is pushing?
Mr. External Mrs. Fields
![Page 9: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/9.jpg)
ENERGY is required to bring the charge up to speed (if it has mass).
ENERGY is required to bring the particle back to rest (if it has mass).
The sum of these two is ZERO.
![Page 10: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/10.jpg)
During this process, who is actually doing work?
Mr. External Mrs. Fields Both of them Neither of them.
![Page 11: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/11.jpg)
Clearly
Both are doing work. BOTH are applying a force through a
distance. BOTH get tired!
![Page 12: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/12.jpg)
About the work that they do .. Mrs. Fields does more work than Mr.
External. Mr. External does more work than Mrs.
Fields. Both do the same amount of work. Each does the negative amount of work
than the other does.
![Page 13: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/13.jpg)
Each does the negative amount of work than the other does.
![Page 14: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/14.jpg)
So, when we move a charge in an Electric Field ..
Move the charge at constant velocity so it is in mechanical equilibrium all the time.
Ignore the acceleration at the beginning because you have to do the same amount of negative work to stop it when you get there.
![Page 15: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/15.jpg)
Summary--Summary--
When an object is moved from one point to another in an Electric Field, It takes energy (work) to move it. This work can be done by an external force (you).
You can also think of this as the FIELDFIELD
doing the negativenegative of this amount of work on the particle.
![Page 16: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/16.jpg)
And also remember:
The net work done by a conservative (field)force on a particle moving
around a closed path is
ZERO!
![Page 17: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/17.jpg)
IMPORTANT
The work necessary for an external agent to move a charge from an initial point to a final point is INDEPENDENT OF THE PATH CHOSEN!
![Page 18: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/18.jpg)
The Electric Field
Is a conservative field. No frictional losses, etc.
Is created by charges. When one (external agent) moves a test
charge from one point in a field to another, the external agent must do work.
This work is equal to the increase in potential energy of the charge.
It is also the NEGATIVE of the work done BY THE FIELD in moving the charge from the same points.
![Page 19: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/19.jpg)
Electric Potential EnergyElectric Potential Energy
When an electrostatic force acts between two or more charged particles, we can assign an ELECTRIC POTENTIAL ENERGY U to the system.
The change in potential energy of a charge is the amount of work that is done by an external force in moving the charge from its initial position to its new position.
It is the negative of the work done by the FIELD in moving the particle from the initial to the final position.
![Page 20: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/20.jpg)
Definition – Potential Energy
PE or U is the work done by an external agent in moving a charge from a REFERENCE POSITION to a different position.
A Reference ZERO is placed at the most convenient position Like the ground level in many gravitational
potential energy problems.
![Page 21: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/21.jpg)
Example:
E
Zero Levelq
F
d
Work by External Agent
Wexternal = F d = qEd= U
Work done by the Fieldis:Wfield= -qEd = -Wexternal
![Page 22: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/22.jpg)
AN IMPORTANT DEFINITION
Just as the ELECTRIC FIELD was defined as the FORCE per UNIT CHARGE:
We define ELECTRICAL POTENTIAL as the POTENTIAL ENERGY PER UNIT CHARGE:
q
FE
q
UV
VECTOR
SCALAR
![Page 23: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/23.jpg)
UNITS OF POTENTIAL
VOLTCoulomb
Joules
q
UV
![Page 24: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/24.jpg)
Let’s move a charge from one point to another via an external force. The external force does
work on the particle. The ELECTRIC FIELD
also does work on the particle.
We move the particle from point i to point f.
The change in kinetic energy is equal to the work done by the applied forces. Assume this is zero for now.
q
WV
WUUU
also
WW
K
if
WWKKK
applied
appliedif
fieldapplied
fieldappliedif
0
![Page 25: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/25.jpg)
Furthermore…
VqW
so
q
W
q
UV
applied
applied
If we move a particle through a potential difference of V, the work from an external
“person” necessary to do this is qV
![Page 26: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/26.jpg)
Consider Two Plates
OOPS …
![Page 27: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/27.jpg)
![Page 28: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/28.jpg)
Look at the path issue
![Page 29: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/29.jpg)
ImportantImportant
We defined an absolute level of potential. To do this, we needed to define a
REFERENCE or ZERO level for potential. For a uniform field, it didn’t matter where we
placed the reference. For POINT CHARGES, we will see shortly
that we must place the level at infinity or the math gets very messy!
![Page 30: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/30.jpg)
An Equipotential Surface is defined as a surface on which the potential is constant.
0VIt takes NO work to move a charged particlebetween two points at the same potential.
The locus of all possible points that require NO WORK to move the charge to is actually a surface.
![Page 31: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/31.jpg)
A collection of points that have the same
potential is known as an equipotential
surface. Four such surfaces are shown in
the figure. The work done by as it moves
a charge be
E
q
Equipotential surfaces
tween two points that have a
potential difference is given by:
V
W q V
2 1
2 1
For path I : 0 because 0
For path II: 0 because 0
For path III:
For path IV:
When a charge is moved on an equipotential surface 0
The work don
I
II
III
IV
W V
W V
W q V q V V
W q V q V V
V
Note :
e by the electric field is zero: 0W
W q V
(24 - 10)
![Page 32: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/32.jpg)
SA B
V
q
F
E
r
Consider the equipotential surface at
potential . A charge is moved
by an electric field from point A
to point B along a path
V q
E
The electric field E is perpendicular
to the equipotential surfaces
.
Points A and B and the path lie on S
r
Lets assume that the electric field forms an angle with the path .
The work done by the electric field is: cos cos
We also know that 0. Thus: cos 0
0, 0,
E r
W F r F r qE r
W qE r
q E r
0 Thus: cos 0
The correct picture is shown in the figure belo
9
w
0
(24 - 11)S
V
E
![Page 33: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/33.jpg)
Field Lines are Perpendicular to the Equipotential Lines
![Page 34: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/34.jpg)
Equipotential
)(0 ifexternal VVqWork
![Page 35: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/35.jpg)
dF W
Keep in Mind
Force and Displacement are VECTORS!
Potential is a SCALAR.
![Page 36: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/36.jpg)
UNITS
1 VOLT = 1 Joule/Coulomb For the electric field, the units of N/C can be
converted to: 1 (N/C) = 1 (N/C) x 1(V/(J/C)) x 1J/(1 NM) Or
1 N/C = 1 V/m So an acceptable unit for the electric field is now
Volts/meter. N/C is still correct as well.
![Page 37: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/37.jpg)
In Atomic Physics
It is sometimes useful to define an energy in eV or electron volts.
One eV is the additional energy that an proton charge would get if it were accelerated through a potential difference of one volt.
1 eV = e x 1V = (1.6 x 10-19C) x 1(J/C) = 1.6 x 10-19 Joules.
![Page 38: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/38.jpg)
Coulomb Stuff: A NEW REFERENCE: INFINITY
204
1
r
qE
Consider a unit charge (+) being brought from infinity to a distance r from a Charge q:
q r
To move a unit test charge from infinity to the point at a distance r from the charge q, the external force must do an amount of work that we now can calculate.
x
![Page 39: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/39.jpg)
Just Do It!
![Page 40: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/40.jpg)
OK, doing it!
AB
r
r
unit
AB
rrkq
r
drkq
dr
qk
dVV
B
A
112
2sr
sE
Set the REFERENCE LEVEL OF POTENTIALat INFINITY so (1/rA)=0.
![Page 41: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/41.jpg)
For point charges
i i
i
r
qV
04
1
![Page 42: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/42.jpg)
r1
r2
r3
P
q1
q3
q2
Consider the group of three point charges shown in the
figure. The potential generated by this group at any
point P is calculated using the principle of super
V
Potential due to a group of point charges
1
31 2
1 2
2 3
31 21 2 3
1 2 3
1 2 3
position
We determine the potentials , and generated
by each charge at point P.
1 1 1, ,
4 4 4
We add the thr
1
ee terms:
1 1
4 4 4
o o
o o o
o
V ,
qq qV
r r
V V
qq qV V V
r r r
V V V V
1.
2.
3r
1 2
11 2
The previous equation can be generalized for charges as
1 1 1 1...
follo
4 4 4 4
ws:n
n i
o o o n o i
q qq q
r r
n
Vr r
(24 - 5)
1 2 3V V V V
![Page 43: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/43.jpg)
For a DISTRIBUTION of charge:
volume r
dqkV
![Page 44: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/44.jpg)
dqO A
Potential created by a line of charge of
length L and uniform linear charge density λ at point P.
Consider the charge element at point A, a
distance from O. From triangle OAP we
dq dx
x
Example :
2 2
2 2
2 20
2 2
2
2
0
2
2
2
2
have:
Here is the distance OP
The potential created by at P is:
1 1
4 4
4
ln4
l
l
n ln4
n
o o
L
o
L
o
o
r d x d
dV dq
dq dxdV
r d x
dxV
d x
V x d x
V L L x
dxx
d
x dd x
(24 - 8)
![Page 45: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/45.jpg)
A
B
VV+dV
The work done by the electric field is given by:
(eqs.1)
also cos cos (eqs.2)
If we compare these two equations we have:
cos cos
From triangle PAB we see that
o
o
o o
W q dV
W Fds Eq ds
dVEq ds q dV E
ds
cos is the
component of along the direction s.
Thus:
s
s
E
E E
VE
s
s
VE
s
(24 - 13)Calculating the electric field E from the potential V
![Page 46: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/46.jpg)
A
B
VV+dV
We have proved that: s
VE
s
s
VE
s
The component of in any direction is the negative of the rate
at which the electric potential changes with distance in this direction
E
If we take s ro be the - , -, and -axes we get:
If we know the function ( , , )
we can determine the components of
and thus the vector itself
x
y
z
x y z
V x y z
E
VE
xV
Ey
VE
z
E
ˆˆ ˆV V VE i j k
x y z
(24 - 14)
![Page 47: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/47.jpg)
Potential energy of a system of point charges
We define as the work required to assemble the
system of charges one by one, bringing each charge
from infinity to its final position
Using the above de
U
U
finition we will prove that for
a system of three point charges is given by:U
2 3 1 31 2
12 23 134 4 4o o o
q q q qq qU
r r r x
y
O
q1
q2
q3
r12
r23
r13
(24 - 15)
, 1
each pair of charges is counted only once
For a system of point charges the potential energy is given by:
Her1
e is the separation between and 4
T
i
ij
ni j
i jo iji j
i j
n q U
r q qq q
Ur
Note :
he summation condition is imposed so that, as in the
case of three point charges, each pair of charges is counted only once
i j
![Page 48: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/48.jpg)
1
1
2
2 2
1 1 22
12 12
3
3 3
1 2
13 23
1 3 23
13
Bring in
0
(no other charges around)
Bring in
(2)
(2)4 4
Bring in
(3)
1(3)
4
1
4
o o
o
o
q
W
q
W q V
q q qV W
r r
q
W q V
q qV
r r
q q qW
r
Step2 :
Step 1 :
e
St p3 :
3
23
1 2 3
2 3 1 31 2
12 23 134 4 4o o o
q q q qq
q
r
W W W W
qW
r r r
(24 - 16)
O
x
y
q1
Step 1
x
y
O
r12q1
q2
Step 2
Step 3
x
y
O
r12
r13
r23
q1
q2
q3
![Page 49: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/49.jpg)
path
A
B
conductor
0E
We shall prove that all the points on a conductor
(either on the surface or inside) have the same
potential
Potential of an isolated conductor
Consider two points A and B on or inside an conductor. The potential difference
between these two points is give by the equation:
We already know that the electrostatic field
B A
B
B A
A
V V
V V E d S
>>>>>>>>>>>>>>
inside a conductor is zero
Thus the integral above vanishes and for any two points
on or inside the conductor.B A
E
V V
A conductor is an equipotential surface
(24 - 17)
![Page 50: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/50.jpg)
We already know that the surface of a conductor
is an equipotential surface. We also know that
the electric field lines are perpendicular to the
equipote
Isolated conductor in an external electric field
ntial surfaces.
From these two statements it follows that the electric field vector is
perpendicular to the conductor surface, as shown in the figure.
All the charges of the conductor reside on the surface and a
E
rrange
themselves in such as way so that the net electric field inside the
conductor
The electric field just out side the conductor is:
0.
in
outo
E
E
(24 - 18)
![Page 51: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/51.jpg)
Electric field and potential
in and around a charged
conductor. A summary
All the charges reside on the conductor surface.
The electric field inside the conductor is zero 0
The electric field just outside the conductor is:
The electric field
in
outo
E
E
1.
2.
3.
4. just outside the conductor is perpendicular
to the conductor surface
All the points on the surface and inside the conductor have the same potential
The conductor is an eequipotential surf
5.
ace
0inE
ˆouto
E n
E
E
n̂
n̂(24 - 19)
![Page 52: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/52.jpg)
In the figure, point P is at the center of the rectangle. With V = 0 at infinity, what is the net electric potential in terms of q/d at P due to the six charged particles?
![Page 53: ELECTRIC POTENTIAL Spring, 2008 Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the.](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bf711a28abf838c7dfa4/html5/thumbnails/53.jpg)
Continuing
dd
s
ddd
dds
12.152
4
5
42
222
222
s
1 2 3
45 6
d
qxV
d
qk
d
qk
d
q
d
qkV
d
qqqq
d
qqk
r
qkV
i i
i
91035.8
93.093.8812.1
108
12.1
5533
2/
22