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Force between Two Point Charges
• The force between two point charges is
– directly proportional to the magnitude of each
charge (q1, q2),
– inversely proportional to the square of the
separation between their centers (r),
– directed along the line connecting their centres.
-q2
+q1
r
Coulomb’s Law
• Coulomb's law describes the force between two
charged particles.
For a vacuum
1
q1q2
F (
) 2
4o r
Where o is called the permittivity
of free space and
o = 8.85 × 10-12 F m-1
And also
1
4o
 9.0 109 N m 2 C -2
Electric Fields
http://www.colorado.edu/physics/2000/applets/nforcefield.html
• The space around a charged body, where electric
force is experienced by a test charge, is called an
electric field.
• By a test charge we
mean a charge so small
that the force it exerts
does not significantly
alter the distribution of
the charges that create
the field.
Electric Field Lines
• The electric field lines indicate the direction of the
force due to the given field on a positive test charge.
• The field points in the direction tangent to the field
line at any point.
• The number of field lines drawn per unit crosssectional area is proportional to the electric field
strength.
F
+q
Properties of Field Lines
http://surendranath.tripod.com/FieldLines/FieldLines.html
• Electric field lines start on positive charges and end on
negative charges.
• The number starting or ending is proportional to the
magnitude of charge.
• The field lines cannot cross.
• The closer the lines the
stronger the field.
• Where the lines are parallel
and uniform spaced, the field
is uniform.
Electric Field Patterns (1)
• Electric field lines for a
single positive point
charge
• Electric field lines for a
single negative point
charge
Electric Field Patterns (2)
• Electric field lines for two
charges of opposite sign.
• Electric field lines for two
equal positive charges
Electric Field Patterns (3)
• Electric field lines between two oppositely
charged parallel plates.
Electric Field Strength
• The electric field strength , E, at any point in an
electric field is defined as the force per unit charge
exerted on a tiny positive test charge at that point.
F
E
q
Unit : N C-1 or V m-1
• E represents a vector quantity whose direction is
that of the force that would be experienced by a
positive test charge.
• The magnitude of q must be small enough not to
affect the distribution of the charges that are
responsible for E.
Electric Field Strength due to a
Point Charge
• By Coulomb’s law
F (
4o
)
Qq
r2
E
F
E
q
• By the definition of E
Then we have
1
1
Q
E(
) 2
4o r
q
r
Q
Notice that E depends only on Q which produces
the field, and not on the value of the test charge q.
Vector Addition of Electric Field
• Suppose we have several point charges Q1,
Q2 and Q3 etc. Then we can
– Evaluate E1, E2 and E3 etc., and
– Find E = Ei by using vector addition.
E1
E
r1
+Q1
r2
E2
-Q2
Electric Field and Conductor
• Any net charge on a good
conductor distributes
itself on the surface.
• E is always perpendicular
to the surface outside of
the conductor. (i.e. E has
no component parallel to
the surface.)
• E is zero within a good
conductor.
If the charge are kept moving, as in current, these properties
need not apply
Electric Field due to a Charged
Spherical Conductor
E
• Inside the sphere
– The electric field is zero. 
• Outside the sphere
o
– For r  a
1 Q
E(
) 2
4o r
• On the surface of the sphere

E
o
a
Where  is the surface charge density.
r
Electric Field due to a Non-conducting
Charged Sphere
• Inside a non-conductor,
which does not have
free electrons, an
electric field can exist.
• The electric field outside
a nonconductor need not
to be perpendicular to
the surface.
E

o
a
r
Electric Potential Energy
• The Coulomb force is a conservative force
(i.e. The work done by it on a particle which
moves around a closed path returning to its
initial position is zero.)
• Therefore, a particle moving under the
influence of a Coulomb force is said to have
an electric potential energy defined by
•
U = qV
Electric Potential Energy of a System
• Consider an electric field formed by a system of
N charges.
• Work has to be done to assemble the charges
from infinity in their final positions.
• The electric potential energy of the field is
defined to be the algebraic sum of the electric
potential energy for every pair of charges.
1
U   qiVi
2 i
Electric Potential
• Electric potential is a measure of the electrical
potential energy per unit charge at a point in
an electric field.
• The electric potential at a point in an electric
field is the work done in moving a unit
positive charge from infinity to that point.
W
V
q
Unit : volts (V)
• Electric potential is a scalar quantity.
Field Strength and Potential Gradient
http://www.falstad.com/vector2de/
• The work done by a force F to move the test charge
q against the electric force by a small distance r is
W  Fr
As
W
V 
q
and
V
r
dV
Hence E  
dr
F
E
q
We get E  
for r 0
i.e. Electric field strength = -potential gradient
Electric Potential due to a Point
Charge
• In terms of the E-field, the electric potential is
defined by
r
V    Edr

The ‘-’ sign indicates that work is done against
the electric force.
• For the electric field due to a point charge Q, it can
be shown that
1 Q
V
4o r
Electric Potential for a Charged
Spherical Conductor
• Inside the sphere the
V
electric potential is
constant, but not zero.
1 Q
• The field at any point 4o a
outside the sphere is
exactly the same as if
the whole charge were
concentrated at the
centre of the sphere.
a
0
1
Q
4o r
r
Zero Potential
• The practical zero potential is that of the
Earth.
• The theoretical zero potential, according to
the definition of V, is that of a point at
infinity.
Potential Difference
• The potential difference across two points A and B
is defined as the work done by the electric field to
move a unit charge from point B to point A.
VAB
W
  VB  VA
q
VB>VA if an external agent does
positive work when moving a
positive charge.
• The work done is independent of path.
Electric Potential between two
Charged Parallel Plates
• The work done by the electric field E to move a
positive charge q from A to B is
• W = qVAB
As W = Fd
and
F = qE
VAB = Ed
Where d is the distance
between AB
d
Equipotentials
• An equipotential surface is one on which all
points are at the same potential.
– The potential difference between any two
points on the surface is zero.
– No work is required to move a charge along an
equipotential.
– The surface of a conductor is an equipotential
surface.
Contours
http://maxwell.ucdavis.edu/~electro/potential/equipotential.html
• The concept of potential, V, in electricity is
equivalent to the concept of altitude, h, in
the case of gravitational field.
Equipotential surfaces and Field Lines
(1)
Equipotential surfaces and Field Lines
(2)
• The equipotentials are
always perpendicular
to the field lines.
• The density of the
equipotentials
represents the strength
of the electric field.
• The equipotentials
never cross each other.
A conducting Material in an Electric Field
• Consider a pair of oppositely charged plates
which established a uniform field between them.
conductor
V/V
+
-
0
E/V m-1
x/m 0
x/m
Electrostatic Shielding
+
-
• The field inside the hollow metal box is zero.
• A conducting box used in this way is an
effective device for shielding delicate
instruments and electronic circuit from
unwanted external electric field.
• The inside of a car or an airplane is relatively
safe from lightning.
Comparison between Electrostatic
and Gravitational Fields
Electrostatic field
Field strength (unit)
Force
Field strength outside
isolated sphere
Potential outside
isolated sphere
Energy transferred
E 
F
q
q1q2
4o r 2
1 Q
E
4o r 2
F
V
1
(N
1 Q
4o r
W=qV
C-1)
Gravitational field
F
g
m
F  G
(N kg-1)
m1m2
r2
M
g  G 2
r
V  G
M
r
W=mV
Differences between Electrostatic
field and Gravitational field
• The gravitational force is always attractive
while the electric force can either be
attractive or repulsive.
• An electric field can be shielded while a
gravitational field cannot.