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ELECTRICITY
PHY1013S
POTENTIAL
Gregor Leigh
[email protected]
PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRIC POTENTIAL
Learning outcomes:
At the end of this chapter you should be able to…
Distinguish carefully between electrical potential energy,
potential difference and potential (and other terminology).
Determine the electric potentials at various points in fields
due to specific charge distributions, and illustrate these
potentials using several graphical representations.
Calculate electric potential from electric field & vice versa.
Apply the law of conservation of energy to determine the
behaviour of charged particles in electric fields.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
GRAVITATIONAL
POTENTIAL DIFFERENCE
Objects have different potential energies at different
points (heights) in a gravitational field.
1m
2 kg
4 kg
80 g
0.5 m
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
GRAVITATIONAL
POTENTIAL DIFFERENCE
The actual difference in potential energy between the two
points depends on the mass being moved.
U = 2  1  9.8 J
2 kg
U = 0.08  1  9.8 J
80 g
4 kg
U = 4  0.5  9.8 J
1m
0.5 m
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
GRAVITATIONAL
POTENTIAL DIFFERENCE
But if instead we consider the difference in the potential
energy per unit of mass (i.e. for each kilogram) between
the two points, we are considering a property of the field.
2 kg
U = 9.8 J per 1 kg
80 g
4 kg
1m
U = 4.9 J per 1 kg
0.5 m
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
GRAVITATIONAL
POTENTIAL DIFFERENCE
We might call this difference in gravitational potential
energy per unit of mass the gravitational
potential difference between the two points: G  U
m
G
U = 9.8 J/kg
J per 1 kg
1m
U = G
4.9 =
J per
4.9 J/kg
1 kg
0.5 m
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
GRAVITATIONAL
POTENTIAL DIFFERENCE
G  U
Units: [J/kg] = [greg, G]
m
And hence we might (in the interests of obfuscation) talk
about the “greggage” between two points in a
gravitational field.
G = 9.8 J/kg
G
1m
GG
= 4.9
= 4.9
J/kg
G
0.5 m
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRIC POTENTIAL ENERGY
Electric potential energy
Electrostatic potential energy
Uelec
But only changes or differences in potential energy are
meaningful. As a field does work on a charged particle
the particle loses potential energy:
U = Uf – Ui = –W
Notes:
The energy is the energy of a system of charges,
but you will hear “the energy of a particle…”.
The work done is done by the force on the particle
due to the other charge(s),
but you will hear “the work done by the field…”.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRIC POTENTIAL DIFFERENCE
The difference in the amount of electric potential energy
per unit of charge between one point and another in an
electric field is known as…
…the potential difference between those points.
Electric potential difference
Potential difference
V   U
q
V  W
q
Units: [J/C = volt, V]
Hence electric potential difference is sometimes
(colloquially) referred to as the voltage between two
points, or “across” a component in a circuit.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRIC POTENTIAL, V
But what could possibly be meant by “the potential at
one point”?
Electric potential (at a point)
Potential
(??!)
V U
q
Using infinity as our reference (zero) point, Ui = 0,
and hence:
or simply:
U = Uf – Ui = Uf – 0 = Uf = –W
U = –W
Hence the potential at a point is given by:
W
V
q
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRIC POTENTIAL, V
Notes:
Electric potential is a property of the field (or,
more specifically, it depends on the source
charges and their geometry). Though we use a
“probe” charge to measure it, like the field itself,
potential exists whether an “intruder” charge is
there to experience it or not.
Electric potential is a scalar quantity.
Like potential difference, potential is measured in
volts (joules/coulomb).
w = qV  [J = C V]… but sometimes it is easier to
use the electron volt (eV): 1 eV = 1.6  10–19 J.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
EQUIPOTENTIAL SURFACES
An equipotential surface is
a collection of points which
all have the same potential.
b
V1 = 80 V
a
c
V2 = 60 V
V3 = 40 V
No net work is done by or
against the electric field
when a charge moves
between two points on the same equipotential surface
(whatever route it follows).
The work done moving a charge from one equipotential
surface to another is independent of the path taken.
Equipotential surfaces lie perpendicular to field lines.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
EQUIPOTENTIAL CONTOURS
Equipotential surfaces (which lie perpendicular to the field
lines) can also be represented as equipotential contours:
+
+
–
equipotential contours
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CALCULATING THE POTENTIAL FROM
THE ELECTRIC FIELD
A particle with charge q moves from
initial position i to final position f
along an arbitrary path in a
non-uniform field…
i
+
q
+
ds
F
f
At any point the force acting on the particle is F  qE
and the differential work done by
the field during a displacement is dW  F  ds  qE  ds
Integrating over the whole path for
the net work done by the field, we get
f
Wif  q  E  ds
i
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CALCULATING THE POTENTIAL FROM
THE ELECTRIC FIELD
i
F  qE
dW  F  ds  qE  ds
q
+
ds
f
Wif  q  E  ds
f
F
i
Wif
And since (from our definition of V) V f  Vi 
q
If i is at infinity (where Vi = 0), then the
potential at any point relative to infinity is
f
V    E  ds
i
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL DUE TO A POINT CHARGE
Letting a test charge move radially
inwards from  to P…
E
dr
q0
ds
E  ds  E cos180 ds   E ds  E dr'
P
r
V    E dr'

q
r
r
1 dr'
V  

4 0  r' 2
q  1 r
V  
 

4 0  r'  
and hence…
r'
V
1 q
4 0 r
+ q
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL DUE TO
MULTIPLE POINT CHARGES
The principle of superposition applies, i.e. V  Vi .
i
Hence:
Notes:
qi
V
4 0 
i ri
1
The sign of each qi determines the sign of its Vi,
but the addition is algebraic, not vector!
For a continuous charge distribution:
V
dq
4 0  r
1
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL DUE TO AN ELECTRIC DIPOLE
P
z
r+
The potential at P is…
V
r
+q
++
r–
–q
4 0r

q
4 0r
V 
q  1 1 
 

4 0  r r 
V 
q  r  r 
4 0  r r 

s O
q
–
When r >> s…
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL DUE TO AN ELECTRIC DIPOLE
P
z
r+
r
+q
++
s O
–q
–
r– r+  r2
r–

qs  p
q  r  r 
V
4 0  r r 
When r >> s…
r– – r+  s cos
V 
s cos
4 0 r 2
V
p cos
4 0 r 2
r– – r+  s cos
q
1
Note: V = 0 for all points in the plane defined by  = 90°
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CALCULATING THE ELECTRIC FIELD
FROM THE POTENTIAL
A positive test charge moves
along the path interval between
two equipotential surfaces.
The potential difference between
the surfaces is dV.
The work done by the field E is dW = –q dV
and also
E
q
+
s

ds
V
V + dV
dW  F  ds  qE  ds  qE cos ds
From which we get
E cos   dV
ds
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CALCULATING THE ELECTRIC FIELD
FROM THE POTENTIAL
… E cos   dV
ds
E cos is simply the component of the
electric field in the direction of ds ,
so we can write
E s   V
s
E
q
+
s

ds
V
V + dV
Taking this direction successively along
the three principle axes, we derive the components of E:
E x   V ; E y   V ; E z   V
x
y
z
And if the electric field is uniform…
E   V
s
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRICAL POTENTIAL ENERGY OF
A SYSTEM OF POINT CHARGES
The electric potential energy of a system of fixed charges
is equal to the work done by an external agent in
assembling the system.
q1
P
to charge q2
+
r
While q2 is still at , the potential at the
position P which will be occupied by q2 is V 
1 q1
4 0 r
Bringing q2 in from  to P, requires work: Wagent  W  q2V
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ELECTRICAL POTENTIAL ENERGY OF
A SYSTEM OF POINT CHARGES
q2
q1
+
+
r
q1
1
Since V 
and
4 0 r
Wagent  q2V
and U  Wagent …
… the potential energy of the pair of charges is thus
q1 q2
1
U
4 0 r
If q1 and q2 are unlike charges the work is done by the
field, and the system has negative potential energy.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL OF A CHARGED
ISOLATED CONDUCTOR
Excess charge on an isolated conductor distributes itself
on the surface of the conductor in such a way that the
field inside the conducting material is zero (regardless of
whether the conductor has an internal cavity – which
may or may not contain a net charge).
f
V    E  ds  0
i
Thus the potential is the same at all points on and inside
the conductor.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
POTENTIAL OF A CHARGED
ISOLATED CONDUCTOR
For a charged spherical conductor (solid or hollow) of
radius r0 …
… And remembering that the
electric field is the derivative
V(r)
of the potential…
E s   V
s
 1r
E(r)


r0

r
r0
1
r2
r
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CHARGE DISTRIBUTION ON A
NON-SPHERICAL CONDUCTOR
The surface of the conductor is an equipotential surface.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CHARGE DISTRIBUTION ON A
NON-SPHERICAL CONDUCTOR
The further one moves away from a tiny conductor, the
more the equipotential surfaces resemble those around a
point charge, i.e. they become spherical.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CHARGE DISTRIBUTION ON A
NON-SPHERICAL CONDUCTOR
In order to “morph” into spheres, equipotential surfaces
near small-radius convex surface elements have to be
closer than they are near “flatter” parts of the surface.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CHARGE DISTRIBUTION ON A
NON-SPHERICAL CONDUCTOR
Equipotential surfaces are closest together where the
electric field is strongest.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
CHARGE DISTRIBUTION ON A
NON-SPHERICAL CONDUCTOR
Thus on an isolated conductor the concentration of
charges and hence the strength of the electric field is
greater near sharp points where the curvature is large.
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PHY1013S
ELECTRICITY
ELECTRIC POTENTIAL
ISOLATED CONDUCTOR IN AN
EXTERNAL ELECTRIC FIELD
For an isolated, uncharged
conductor in an external field,
the free charges (electrons)
distribute themselves on the
surface of the conductor in
such a way that …
E=0
the net field inside the conducting material is zero;
the net electric field at the surface is perpendicular to the
surface.
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