Download Electrical Potential Energy & Electrical Potential

Electrical Potential Energy & Electrical
Potential

Work, Potential Energy and the Electric Force

Electrical Potential

Equipotential Surfaces

Dielectrics, Capacitors, and Energy
Electrical Potential Energy & Electrical
Potential
Work,
Potential Energy and the
Electric Force
Electrical Potential
 Equipotential Surfaces
 Dielectrics, Capacitors, and Energy

Work, Potential Energy, and the Electric Force
Consider a positive charge q, moving west through a
displacement S at a constant velocity in a region occupied by a
uniform eastward electric field E.
S
+q
E
Work, Potential Energy, and the Electric Force
The charge experiences an eastward electric force. For it to move
at constant velocity, an equal westward force is required.
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
Work done by the westward force:
Wwest  qES cos 0  qES
Work done by the electrical force:
Welec  qES cos180  qES
Net work = D kinetic energy = zero
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
The electric force is conservative. Its work does not change the
energy of the object.
The westward force is nonconservative. Its work changes the
total energy: DE  W
 qES
west
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
The increase, qES, in the energy of the object is
electrical potential energy (EPE).
Like any other form of energy, it has SI units of joules (J).
Every conservative force is associated with a form of potential
energy.
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
The work done by the forces is the same, even if we go from the
starting to the ending point by a meandering path. Work done
by a conservative force is path-independent.
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
The increase in electrical potential energy (EPE) is also
unaffected by our choice of path. D EPE depends only on E,
q, and the endpoint locations.
S
F
+q
F
E
Work, Potential Energy, and the Electric Force
Call the endpoints of our travel A (beginning) and B (end).
Travel from A to B with a “test charge” q0.
WAB  DEPEAB  EPEB  EPEA
Divide through by q0 :
WAB DEPEAB EPEB EPEA



q0
q0
q0
q0
Work, Potential Energy, and the Electric Force
WAB DEPEAB EPEB EPEA



q0
q0
q0
q0
Define a new quantity, electrical potential:
and
WAB
 DVAB  VB  VA
q0
SI units of electrical potential:
W
V
q0
joule
 volt (V)
coulomb
Electrical Potential Energy & Electrical
Potential

Work, Potential Energy and the Electric Force
Electrical
Potential
Equipotential Surfaces
 Dielectrics, Capacitors, and Energy

Electrical Potential
WAB
 DVAB  VB  VA
q0
Since we’ve talked about the electrical potential of points A
and B, let’s look at what that means. Consider a positive
point charge, Q, and a point A that is a distance r from Q:
Y
r
X
+Q
point A
x=r
Electrical Potential
What is the electrical potential at point A?
Y
r
X
+Q
point A
x=r
To answer that question, we need to know how much
work is required to bring a test charge from a point
infinitely distant from Q, to point A.
Electrical Potential
Y
r
X
+Q
point A
x=r
The force required to move the test charge at a
constant velocity is not constant with the distance x
q0 Q
from Q:
F  k
x
2
In fact, the force isn’t even linear with x … so we
can’t calculate and use an average force.
Electrical Potential
Y
r
X
+Q
point A
x=r
Calculating the potential at point A is a calculus
problem. The work required to move the test charge
an infinitesimal distance dx is dW:
q0 Q
dW  Fdx  k 2 dx
x
Electrical Potential
Integrate to calculate W:
q0Q
dW  Fdx  k 2 dx
x
r 1
 1 1  kq0Q
W  kq0Q  2 dx   kq0Q   
 x
r
r 
W kQ
V

q0
r
Electrical Potential
The potential of a point is relative to zero potential,
which is located infinitely far away.
When more than one charge is present, the potential is
the algebraic sum of the potentials due to each of
the charges, individually.
“Electric potential” is not the same thing as “electrical
potential energy.”
Electrical Potential
Gravitational
Electrical
W  force  distance  mgh
W
 gh
m
W  force  distance  qE  S
W
 V  ES
q
Another way to express the magnitude of the electric
field: as a “potential gradient:”
V
E
S
(volts/met er)
Electrical Potential Energy & Electrical
Potential
Work, Potential Energy and the Electric Force
 Electrical Potential

Equipotential

Surfaces
Dielectrics, Capacitors, and Energy
Equipotential Surfaces
Equipotential surface: a collection of points that all
have the same electrical potential.
Equipotential Surfaces
The electric field vector is everywhere perpendicular
to equipotential surfaces. Why?
Equipotential Surfaces
The electric force does no work on charges moving on
an equipotential surface. Why?
Electrical Potential Energy & Electrical
Potential
Work, Potential Energy and the Electric Force
 Electrical Potential
 Equipotential Surfaces

Dielectrics,
Capacitors, and Energy
Dielectrics, Capacitors, and Energy
Consider a parallel-plate capacitor, with a charge q
and a plate area A, the plates separated by a
q
distance d. The internal electric field:
E
0 A
A constant electric field E, and a distance d between
the plates, gives the potential difference between
the plates:
qd
V  Ed 
0 A
Dielectrics, Capacitors, and Energy
qd
V  Ed 
0 A
Solve for the ratio of the charge to the voltage, and
define a new quantity, capacitance:
q 0 A
C 
V
d
SI units: coulomb/volt = C2/J = farad (F)
Dielectrics, Capacitors, and
Energy
Dielectric material: fancy term for
an insulating material.
Separation of surface charge causes
a reduction of the internal electric
field by a factor of k, the
dielectric constant:
E0
k
E
Dielectrics, Capacitors, and Energy
Fill the capacitor with a dielectric:
E
q
k 0 A
qd
V  Ed 
k 0 A
q k 0 A
C 
V
d
Its capacitance is increased by a factor of k, the
dielectric constant.
Dielectrics, Capacitors, and Energy
Springs
Capacitors
Work is required to
compress a spring
Work is required to
charge a capacitor
Work per unit length
increases linearly
with compression
Energy:
Work per unit charge
increases with the
amount of charge
Energy:
1 2
E  kx
2
1
E  CV 2
2