Electric Potential
&
it’s applications
Contents :


Electric Potential
Energy Conservation



The Electric Potential of Point Charges
Equipotential Surfaces
Electric potential due to charged spherical
shell




Liquid flows from higher level to lower level
Electric charge flows from a point of higher
potential to a point of lower potential
Electric field and electric potential are
related to each other.
Electric potential is a scalar quantity.
Electric Potential and Electric Potential Energy
 To move an object in a gravitational field requires work. There
is a change in the object’s Potential Energy.
 To move a charge in an Electric field requires work. There
is a change in the charge’s EPE.
 In gravitational fields a force is required to move masses
apart. i.e positive work is done (by an external agent).
 In electric fields a force is required to push like charges
together. That is positive work is done (by an external agent).
Change in electric potential energy
EPE Increases
PE Increases
Electric field uniform
Work Done by an Electric Force, W=-qoEd
Work Done by an YOU, -W=qoEd
U = UB-UA = -W = q0Ed
Change in electric potential energy
of a charge on moving from A to B
In the above: You do positive work,
the field does negative work.
When things happen spontaneously:
The field does positive work YOU do negative work
Gravitational
field uniform
Change in electric potential
The change in potential energy or the work done depends on the size of the
charge being moved.
V = VB-VA = U/ q0 = -W / q0
Potential difference
between points A to B
Measured in Volts
1 Volt = 1 J/C
Work Done By an Electric FORCE
It’s important to know who does the work!
Electrical potential difference more important
than the actual value of electric potential
For a uniform field between two plates
V = U / q0 = q0Ed / q0 = Ed
or E =- V / s
Potential difference depends only on the plates and NOT on any charge being moved.
The work done depends on the size of the charge being moved.
Electric field and electric potential
As a charge q0 moves in the direction of the electric field, E, the
electric potential, V, decreases.
In particular, if the charge moves a distance <s, the electric
potential decreases by the amount
<V = -E < s.
How is electric field related to electric potential?
Use Dimensional or unit analysis
Electric field is measured in units of N/C
F=ma
WD=Fd
N (kgms2 )

C
C
(kgms2 )m  J
N J V C
 
mC m So
E
V
d
Problem 11
Problem 11:
A
B
C
D
A uniform electric field of 1200 N/C points to the left as shown.
What is the difference in potential (VB-VA) between points B and A?
What is the difference in potential (VB-VC) between points B and C?
What is the difference in potential (VC-VA) between points C and A?
Is it possible to determine the value of the electric potential at point A? If not, explain why.
11.a
b
c
V=VB -VA = 0 because VB =VA
N
)(0.040 m)=-48V
C
N
V=VC -VA = Ed=(1200 )(0.040 m)=+48V
C
V=VB -VC = -Ed=-(1200
Integration
Divide shell into rings of charge, each delimited by the angle  and
the angle +d
Use polar coordinates (r, ,).
d
R
Distance from center: d=(r-Rcos)
Rsin
Surface area of ring:

Rd
Rcos
2 (R sin )
d
r
1
(dQ)d
2 Rsin Rd
dE 
dQ

Q




4 0 (d 2  (R sin )2 )3/ 2
4 R 2
Q
1
(r  R cos )
E
sind

3/2
4 0 2 0 (r  Rcos ) 2  (R sin )2 


A Solid Sphere of Charge
Inside a solid sphere of charge:
1 Q

r̂
E
2
4 0 r
Q  Q
(volume of inner shells)
(volume of sphere)
3
3
r
4
/
3


r
Q
Q  Q
4 / 3   R3
R3
1 Qr
r̂
E
3
4 0 R
for r<R
Why is E~r?
 Onsurface:
1 QR
1 Q
r̂
E
rˆ
3
2
4 0 R
4 0 R
R
r
E
Electric Potential
Equipotentials and Energy
U  qV
Electrical Potential
Wa  b = work done by force in going from a to b along path.
b
b
a
a
Wa b  ∫ F  dl  ∫ qE dl
F
b
U  U b U a  Wa b  ∫ qE  dl
U = potential energy
V  Vb Va 

U b U a
q
q


a
a
U
b
Wa b
q
dl
b
  ∫ E dl
a
•Potential difference is minus the work done per unit
charge by the electric field as the charge moves from a to b.
• Only changes in V are important; can choose the zero at any point.
Let Va = 0 at a = infinity and Vb  V, then:
r
V   ∫ E  dl

V = electric potential
allows us to calculate V
everywhere if we know E
Potential from charged spherical shell
• E-field (from Gauss' Law)
•
r < R : Er = 0
•
r >R:
V
q
4 r
q
4 R
1
q
Er =
4 0 r 2
R
R
• Potential
R
• r > R:
r
r
r
E  dl   ∫ E r ( dr ) 
rr
V(r)
∫
r 

1
4
0
q
Q
r
• r < R:
aR
r
r
r r
q
1 Q
0
V( r )   ∫ E dl   ∫ E r( dr )   ∫E r ( dr ) ∫ E r (dr ) 
40 aR
a
r


rr
R
r
ELECTRIC POTENTIAL for Charged Sphere (Y&F, ex.23.8)
Suppose we have a charged
metal sphere with charge q.
What is the electric potential
as a function of radius r?
N.B. V is continuous but
E is not.
q
Electrical Potential
Two ways to find V at any point in space:
r
r r
• Use electric field:
V   ∫ E dl
• Sum or Integrate over charges:
q1
q2
r2
P
r1
r3
q3
dq
r

V
1 qi
∑ 4 r
i
0 i
P
1 dq
V
∫
4 0 r
Examples of integrating over a distribution of charge:
• line of charge (review this one)
• ring of charge
• disk of charge
You should be able to do these.
Infinite line charge or conducting cylinder.
r
Linear charge density 

E
20r
Va Vb 
b
∫
a
Suppose we set rb
to infinity,
potential is infinite
Edr  ∫
rb


ln( )

ra
2 0r 2 0
Instead, set ra=r and rb=r0 at
some fixed radius r0.
Potential from a charged sphere
V(r) 
1 q
40 r
(where V() 0 )
Er
Equipotential
•
•
•
•
The electric field of the charged sphere has spherical symmetry.
The potential depends only on the distance from the center of the
sphere, as is expected from spherical symmetry.
Therefore, the potential is constant on a sphere which is
concentric with the charged sphere. These surfaces are called
equipotentials.
Notice that the electric field is perpendicular to the equipotential
surface at all points.
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 electric field is always perpendicular
to an equipotential surface!
r r
VB V A  ∫E  dl
B
Why??
A
Along the surface, there is NO change in V (it’s an equipotential!)
Therefore,
r
r
 ∫E  dl  V 0
B
A
We can conclude then, that E  dl is zero.
If the dot product of the field vector and the displacement vector is zero,
then these two vectors are perpendicular, or the electric field is always
perpendicular to the equipotential surface.
EXAMPLES of Equipotential Lines
How to obtain vector E field from V
r
• Determine V from E:
V   ∫ E dl

Example: V due to spherical charge distribution.
Determining E from V:
b
b
a
a
V  Vb Va   ∫ E  dl  ∫ dV
For an infinitesimal step:
dV  E  dl  Edl cos
Cases:
•  = 0:
•  = 90o:
•  = 180o:
Can write:
E

dl
directional derivative
dV = E dl
(maximum)
dV = 0
dV depends on direction
dV = -E dl
dV  E  dl  (Exiˆ  E y ˆj Ez kˆ) (dxiˆ  dy ˆj dzkˆ)
 (Ex dx  E y dy  Ez dz)
Potential Gradient
Take step in x direction: (dy = dz = 0)
Similarly:
And:
dV   (E x dx  E y dy  Ez dz)  E x dx
V
dV

E x 
partial derivative
dx y, z const.
x
Ey  
V
y
Ez  
V
z
V ˆ V ˆ V ˆ
i
j
k)  V
x
y
z

 ˆ  ˆ
gradient operator
 ( iˆ 
j  k)
x
y
z
E  (
Gradient of V points in the direction that V increases the fastest
with respect to a change in x, y, and z.
E points in the direction that V decreases the fastest.
E perpendicular to equilpotential lines.
Potential Gradient
Example: charge in uniform E field
E
U = qEy
q
V = U/q = Ey
where V is taken as 0 at y = 0.
E  V  (  iˆ   jˆ   k)ˆ Ey
x
y
z
 (0iˆ  E ˆj 0kˆ)  E ˆj
Given E or V in some region of space,
can find the other.
y
o
Cylindrical and spherical symmetry cases: Example: E of point charge:
q
For E radial case and r is distance
V

)
Er  
 (
from point (spherical) or axis
r
r 4 0r
(cylindridal):
Er  
V
r
 (
q
4 0
)(
1 q
)
r2
40r 2
If we know the electric field E everywhere,
W AB
VB  V A 
q0
r
r
V B  V A  ∫ E  dl
B
⇒
A
allows us to calculate the potential function V everywhere (keep
in mind, we often define VA = 0 at some convenient place)
If we know the potential function V everywhere,
E  V
allows us to calculate the electric field E everywhere
• Units for Potential! 1 Joule/Coul = 1 VOLT
(remember we measure potential differences with a voltmeter)
 Electric potential is a analogous to height in mechanics and temperature in
heat.
 Electric potential is a scalar quantity.
 It is the negative line integral of electric field between infinity and the given
point.
 Electric potential at any point varies inversely as the distance of the point from
source charge.
 Electric potential at any point due to a positive charge is positive while that due
to a negative charge is negative.
 Electric Potential Energy is not the same as Electrical Potential.
 Electrical Potential can also be described by the terms, potential difference,
voltage, potential drop, potential rise, electromotive force, and EMF. These
terms may differ slightly in meaning depending on the situation.
 The variable we use for potential difference is V and the unit for potential
difference is also V (volts). Don’t let that confuse you when you see V = 1.5V
 The electron volt is not a smaller unit of the volt, it’s a smaller unit of the Joule.