Post on 18-Dec-2015
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
E
+ q
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
E
+ q
F
F
Work, Potential Energy, and the Electric Force
Work done by the westward force:
Work done by the electrical force:
Net work = kinetic energy = zero
S
E
+ q
F
F
qESqESWwest 0cos
qESqESWelec 180cos
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:
S
E
+ q
F
F
qESWE west
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
E
+ q
F
F
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
E
+ q
F
F
Work, Potential Energy, and the Electric Force
The increase in electrical potential energy (EPE) is also unaffected by our choice of path. EPE depends only on E, q, and the endpoint locations.
S
E
+ q
F
F
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.
Divide through by q0 :
ABABAB EPEEPEEPEW
0000 q
EPE
q
EPE
q
EPE
q
W ABABAB
Work, Potential Energy, and the Electric Force
Define a new quantity, electrical potential:
and
SI units of electrical potential:
0000 q
EPE
q
EPE
q
EPE
q
W ABABAB
0q
WV
ABABAB VVVq
W
0
(V) volt coulomb
joule
Electrical Potential Energy & Electrical Potential
Work, Potential Energy and the Electric Force
Electrical Potential
Equipotential Surfaces Dielectrics, Capacitors, and Energy
Electrical Potential
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:
ABABAB VVVq
W
0
r
Y
X
x = r
+ Qpoint A
Electrical Potential
What is the electrical potential at point A?
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.
r
Y
X
x = r
+ Qpoint A
Electrical Potential
The force required to move the test charge at a constant velocity is not constant with the distance x from Q:
In fact, the force isn’t even linear with x … so we can’t calculate and use an average force.
r
Y
X
x = r
+ Qpoint A
20
x
QqkF
Electrical Potential
Calculating the potential at point A is a calculus problem. The work required to move the test charge an infinitesimal distance dx is dW:
r
Y
X
x = r
+ Qpoint A
dxx
QqkFdxdW
20
Electrical Potential
Integrate to calculate W:
r
kQ
q
WV
r
Qkq
rQkqdx
xQkqW
dxx
QqkFdxdW
r
0
0020
20
111
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
Another way to express the magnitude of the electric field: as a “potential gradient:”
ghm
W
mghW
distanceforce
ESVq
W
SqEW
distanceforce
er)(volts/met S
VE
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 distance d. The internal electric field:
A constant electric field E, and a distance d between the plates, gives the potential difference between the plates:
A
qE
0
A
qdEdV
0
Dielectrics, Capacitors, and Energy
Solve for the ratio of the charge to the voltage, and define a new quantity, capacitance:
SI units: coulomb/volt = C2/J = farad (F)
A
qdEdV
0
d
A
V
qC 0
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 , the dielectric constant:
E
E0
Dielectrics, Capacitors, and Energy
Fill the capacitor with a dielectric:
Its capacitance is increased by a factor of , the dielectric constant.
d
A
V
qC 0
A
qE
0
A
qdEdV
0
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
Work per unit charge increases with the amount of charge
Energy: Energy:
2
2
1kxE 2
2
1CVE