General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
The Electric Potential, Electric Potential Energy and
Energy Conservation
Definition:
The electric potential is electric potential energy per charge (like
we derived the electric field).
Electric Potential Energy U is the energy of a charged object
in an external electric field (Unit Joule J)
V = U/q0
In electricity, it is usually more convenient to use the
Electric Potential V (or Voltage) is electric potential energy per unit
charge, measured in joules per coulomb (Unit Volt V)
The change in electric potential energy per charge is known as
potential difference or voltage and is measured in Volt. Only
changes of electric potential are meaningful.
A charged particle q exerts an electric force (a conservative force)
in an electric field. If the charge q is not static then work is done in
an electric field by moving the charge q. The work is proportional
to the magnitude of charge (More work is needed to move a bigger
charge). This is similar to lift/drop an object against/with gravity.
∆V = ∆U/q0 = -W/q0
1V [Volt] =1 Nm/C
When we dealing with small charges, there is another unit for
energy: electron volt = energy of one electron possesses when it
has moved through a potential difference of 1 Volt
1 eV = 1.602 10-19 J [Joule]
Connection between electric field and electric
potential (E=const.)
∆V = ∆U/q0 = -W/q0 = q0Ed/q0 = Ed
F=q0E
∆U=Eel=qEd
(W=Fs cosθ and ∆U=-W)
(when E-field is homogeneous, E is constant
vector everywhere)
E = ∆V/d=∆V/∆s
If E=const. The change in the
potential per distance is constant
F=mg
∆U=Epot =mgh
Example: Parallel Plate Capacitor
Electric Potential Energy
Gravitational Potential Energy
The electric potential energy increase/decrease in the same
way as like the gravitational energy.
Only differences in potential energy are important
1
For calculating physical quantities it is
the difference in potential which has
significance, not the potential itself.
Therefore, we may choose any arbitrary
point as having zero potential
2
General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
Energy conservation
When we move a charge in an electric field the total energy – the
electric force is conservative – must be conserved.
The Electric Potential of Point Charges
q1q2
2
Electric force F21 = k
r12
If we release test charge it will move (accelerate) away
(Ekin + Eel)initial = (Ekin+Eel)final
At point B: UA-UB=(qEd)A-(qEd)B
(½ mv2+U)initial=(½ mv2+U)final
= (Fd)A-(Fd)B= kq0q/rA- kq0q/rB
(½ mv2+qV)initial=(½ mv2+qV)final
½ mv2final=(½ mv2)initial + q(Vfinal-Vinitial)
or VA-VB= (UA-UB)/ q0= kq/rA- kq/rB
v2final=v2initial+ 2q(∆V)/m
Definition:
Rules:
positives charges accelerate in the direction of decreasing
electrical potential
negative charges in the opposite direction
In both cases the charge moves to a region with a lower
potential energy
if r
∞ then VB=0
The electric potential
for a point charge
kq
V=
[V ]
r
The electric potential
energy for a point
charge
qq
U = q0V = k 0
[J ]
r
3
4
General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
Superposition of two or more point charges
The potential at a given point in space due to several charges can
be found by adding the potential from each charge
VA = k
Another way to draw electric potentials: Contour Maps
q1
q
q
+ k 2 + k 3 + ...
r1
r2
r3
The denser the lines the more great is the change in the
potential (Equipotential surfaces, Equipotential lines)
The electric field points always in the direction of the
decreasing lines
No work is required to move a charge along an equipotential
surface, the electric field at every point on an equipotential
surface is perpendicular to the surface.
5
6
General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
If both spheres have the same potential
The ideal conductor
Large sphere:
Every point on or within such a
conductor is at the same potential.
Ideal conductors are equipotential
objects.
kσ 4πR 2
= kσ 4πR
R
kσ 4πR 2
E=
= kσ 4π
R2
V=
Small sphere:
R2
4 = k ? 4π R
V=
R
2
2
Therefore: ?=σsmall sphere= 2σ
k ? 4π
The electric field is stronger where
the conductor is sharply curved.
Why?
Let’s assume we have to charged
metal spheres both with the same
electric potential.
higher charge density
2
E=
k 2σ 4π
R
4
2
R
4 = k 2σ 4π
The potential of a metal sphere is
given by
kQ
V=
R
The electric field is given by
kQ
E= 2
R
Q is the charge distributed uniformly
over the surface A with a certain charge
density σ
Q= σ A = σ 4πR2
7
8
General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
Dielectrics
Capacitors
A way to increase the capacitance is with an insulting material
between the plates – a dielectric
A capacitor is a device that stores energy
associated with a configuration of charges.
In general, a capacitor consists of two
conductors, one with a charge +Q and the
other with a charge -Q.
Dielectric slab
Capacity is the ability to store charge and energy.
The capacitance C is defined as the ratio of the magnitude of the charge
on either conductor to the magnitude of the potential difference
between the conductors:
Q=CV
or C = Q/V
C: Capacitance [F = Farad]
Field lines terminate on negative charge and start at positive charge
on polarized slab. Less field lines are going through the entire
capacitor.
We reduce E-field by a material dependent factor – the dielectric
constant κ
The parallel-plate capacitor
We can calculated according to Gauss law for a parallel plate
capacitor the electric field as
Q
σ A Q
E= = =
ε0 ε0 ε0 A
V=
C=
Qd
ε0 A
E=
E0
Vacuum κ=1; H2O= 80.4
κ
We know
V = Ed =
C =
We can calculate the potential as V=Ed
E0
κ
d=
V0
κ
Q
Q
Q
=
= κ
= κ C0
V0
V
V0
V decrease with increasing κ
C increase with increasing κ
κ
ε A
Q
= 0
Qd
d
ε0 A
We can expand the equation for the parallel capacitor:
C=
Capacitance of a parallel-plate capacitor
9
ε0 A
d
κ
10
General Physics - PH202 – Winter 2006 – Bjoern Seipel
General Physics - PH202 – Winter 2006 – Bjoern Seipel
Electrical Energy Storage
=> Capacitance dependence also on the
surface area A of the plates and the
distance d. That how a keyboard works
Situation: Parallel plate capacitor with a charge Q on its plates
and potential difference of V. Now we transfer small packages of
charge from one play to the other. The change in electric potential
energy would be ∆U = ∆Q V
Attention! V is not constant. If we would transfer another charge package
we have to calculate with another V
A dielectric breakdown appears when the Electric field is too
strong. That means strong enough to tear atoms apart.
U= QVav = ½ QV
U = ½ CV2
U = ½ Q2/C
Substance
Dielectric strength (V/m)
With Q = CV
V = Q/C
Mica
100 x 106
For a parallel plate capacitor Q = ε0 E A and V = Ed
Teflon
60 x 106
Therefore U = ½ Q V = ½ (ε0 E A) (Ed)= ½ ε0 E2 A d
Paper
16 x 106
Pyrex glass
14 x 106
Neoprene rubber
12 x 106
Air
3.0 x 106
A⋅d is the volume between the plates so we can define a density
(i.e. a quantity divided by its volume)
Electric energy density
UE = electric energy/ volume = ½ ε0 E2
Next time you walk across a carpet and get a shock from the
doorknob think about the fact that you just have produced an
electric field of roughly 3 million V/m.
11
12