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Radial Electric Fields
Any 2 charges exert a force on one another, attractive if they are
opposite and repulsive if they are like charges. Around any charge we can
imagine an invisible field of force (similar to gravity fields), which is an area
of space in which a charge experiences a force.
Around a Positive Point Charge
Field lines
+
Equipotentials
Around a Negative Point Charge
Field lines indicate the
direction of the force
acting on a +ve charge
placed in the field.
Field strength, hence
force increases as you
move closer to the
charge - field lines are
closer together.
-
Charles Augustin
COULOMB
(1736 - 1806)
Physicien français
Q1
Opposite
charges
attract
In 1875 French scientist Coulomb measured the
force acting on two charged spheres and came up
with a fundemental force law which states, “the
electrical force F between two point charges Q1
and Q2 a distance r apart is proportional to the
product of the charges and is inversely
proportional to the square of the distance apart”.
Q2
F
F
r
Q1
Like
F
charges
repel
F  Q1Q2
r2
Q2
F
r
 F =
1 . Q1Q2
4o r2
1
= constant of proportionality
(o is the permittivity of free
space = 8.85x10-12 C2N-1m-2)
1
=
4o
4o
?????? Units?
Notice that the force can be attractive or repulsive depending on the sign of the
charges. If the signs of Q1 and Q2 are inserted in the Coulomb force equation then:
when F is +ve repulsion occurs
when F is –ve attraction occurs
Example
Two charged metal spheres are placed and held on a level wooden table such
that their distances are 20cm apart. Each sphere experiences a repulsive
force of 18μN. One of the sphere holds a charge of -10nC, what is the
charge on the other?
F
-10nC
?
F
20cm
Calculating a Resultant Electrical Field Strength E
Electrical field strength E is a vector quantity and needs to be treated as such when
calculating it’s resultant value at a point when more than one charge is considered. It
is easier not to include the signs of the charges, just consider the direction of E due
to each charge.
HINT Work through problems on the board adding field strength arrows as
you go!
Harder Example - Four point charges are arranged symmetrically as shown in the
diagram below. Calculate the resultant field strength at the centre (marked O).
1
-8nC
+5nC
O
+10nC
4o
2m
2m
+8nC
2m
2m
= 9 x 109 Nm2C-2
Electrical Field Strength E
We define electrical field strength E as “the force acting on unit +ve
charge placed at a point in the field” and is measured in NC-1.
F =
since
1 . Q1Q2
4o
r2
If we make Q1 = Q (charge setting
up field) and Q2 = +1 C, we get:
If Q is a positive point charge
E (NC-1)
+
0
E =
1 . Q
4o r2
If Q is negative point charge
0
E1
r2
r (m)
-
r (m)
Said to create a repulsive field
E (NC-1)
Said to create an attractive field
(like a G-field)
Electrical Potential Ve
The electrical potential Ve at a certain position in an electric field is
defined as, “the work done bringing unit +ve charge from infinity to that
point”. It is measured in JC-1 or volts (V) and is given by:
Ve =
1 . Q
4o r
where Q is the charge setting up the
field (sign needs to be included) and r is
the distance from the centre of Q
Easy way to see where this equation comes from:
E =
1 . Q
4o r2
Work done per unit = force per unit x distance
+ve charge (Ve)
+ve charge
 Ve
=
Potential is essentially the electrical potential
energy that a +1 C charge would have at
that point in the field.
E r
There should be a –ve
here to indicate
increasing potential
opposes the field
direction.
Electrical Potential Ve
How the potential Ve varies within the field depends on the size and sign of the charge
setting up the field, but again potential is taken as zero at infinity. Occasionally to
simplify problems we take the Earth’s potential as zero (‘Earthing’) – see uniform field
notes.
If Q is a positive point charge
Ve (JC-1)
+
If Q is negative point charge
0
Potential
‘Hill’
0
r (m)
V
Gradient = re = - E
Ve rises as you move towards +Q since
work would have to be done move +1 C
inwards overcoming the repulsive
forces. Hence all potentials are
positive
r (m)
Potential
‘Well’
Ve (JC-1)
Gradient =
Ve
= -E
r
Ve decreases as you move towards -Q
since the charge would do the work
for us in attracting a +1 C charge.
Hence all potentials are negative.
Calculating a Net Potential Ve
Electrical Potential Ve is a scalar quantity and needs to be treated as such when
calculating it’s net value at a point when more than one charge is considered. The signs
of the charges must be included.
Example - Four point charges are arranged symmetrically as shown in the diagram
below. Calculate the net potential at the centre (marked O).
1
= 9 x 109 Nm2C-2
4o
-8nC
+5nC
O
+10nC
2m
2m
+8nC
2m
2m
Electrical Potential Energy EPE
As a +ve charge is moved closer to another +ve charge its EPE rises (like pushing it up a
‘potential hill’), whereas it’s EPE would decrease if it moved closer to a –ve charge
(falling down a ‘potential well’).
The EPE of a system of two charges Q1 and Q2 separated by a distance r and is given by:
EPE = 1 .Q1Q2
4o r
If the charges are opposite, EPE is –ve and attraction is
occurring, like e- in atoms.
If they are like charges, EPE is +ve and repulsion is
occurring.
Electrical Potential Difference Ve and Energy Changes
When a charge is moved between two points in an electrical field the work done (energy
change) is independent of the path taken and depends only on the change in potential
(Ve) it experiences.
Electrical pd (Ve) between two points in a field is defined as, “the work done moving
+1 C between the two points.”
Change in energy (work done) when a charge = EPE = qVe
q is moved between the two points
Notes:
1. Change () = final – initial
2. Consideration of the charges involved allows us to establish if EPE
has increased or decreased.
Example
1
4o
= 9 x 109 Nm2C-2
Two charged metal spheres X and Y are placed and held on a level wooden
table as shown below. Calculate:
-10nC
X
a)
the EPE of the system
b)
the potential at X due to Y
c)
the change in potential of X if it is moved 5cm to the left
d)
the change in EPE of the system
+6nC
Y
20cm
Using Electrical Field Strength Graphs
Field Strength around a
positive point charge
Field Strength around a
negative point charge
r1
E (NC-1)
r2
r (m)
r1
r2
r (m)
E (NC-1)
In both graphs the area shaded represents the potential
difference between two positions r1 and r2.
Area is essentially force per unit +ve charge times
distance, which is the work done per unit +ve charge or pd.