Download Electric Potential and Energy

Physics 201
Professor P. Q. Hung
311B, Physics Building
Physics 201 – p. 1/2
Electric Potential and Energy
Summary of last lecture
Electric Potential for a constant electric field:
~ s
VB − VA = −E.~
Physics 201 – p. 2/2
Electric Potential and Energy
Summary of last lecture
Electric Potential for a constant electric field:
~ s
VB − VA = −E.~
Electric Potential for a point charge:
V = k rq
Physics 201 – p. 2/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Example
A parallel plate capacitor is composed on two
charged plates, one positive and the other one
negative. A constant electric field points from the
positive plate to the negative plate. Suppose the
potential difference between the 2 plates is 64 V
and that they are separated by 0.032m. By this
we mean ∆V = V− − V+ = −64V . Between the
two plates, one can draw equipotential planes
parallel to the plates. What is the separation
between 2 of such planes if 3.0 V ?
Physics 201 – p. 3/2
Electric Potential and Energy
Equipotential surfaces and Electric fields
Equipotential Surface:
Surface where the electric potential is the
same everywhere.
Physics 201 – p. 4/2
Electric Potential and Energy
Equipotential surfaces and Electric fields
Equipotential Surface:
Surface where the electric potential is the
same everywhere.
What are the equipotential surfaces around a
point charge?
Physics 201 – p. 4/2
Electric Potential and Energy
Equipotential surfaces and Electric fields
Equipotential Surface:
Surface where the electric potential is the
same everywhere.
What are the equipotential surfaces around a
point charge?
What are the equipotential surfaces between
two charged parallel plates (one + and one -)
(parallel-plate capacitor)?
Physics 201 – p. 4/2
Electric Potential and Energy
Equipotential surfaces for a point charge
V = k qr
Physics 201 – p. 5/2
Electric Potential and Energy
Equipotential surfaces for two point charges
V = k qr
Physics 201 – p. 6/2
Electric Potential and Energy
Equipotential surfaces for a parallel-plate
capacitor
~ =
Recall Active Example 19.3: |E|
electric field.
σ
ǫ0 :
Constant
Physics 201 – p. 7/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Meanings
When a particle moves from one point to
another point on a equipotential surface, the
net electric force does no work. Since
VB − VA = −WAB /q, “equipotential” means
that VB = VA ⇒ WAB = 0.
Physics 201 – p. 8/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Meanings
The electric field always point in the direction
perpendicular to the equipotential surface. If
not, it would have a component parallel to the
surface which means that the surface is no
longer equipotential since the electric field
points in the direction of decreasing potential
Physics 201 – p. 9/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Meanings
A conductor’s surface is an equipotential
surface because the electric field has to be
perpendicular to the conductor’s surface.
Since there cannot be an electric field inside
the conductor, the electric potential inside has
the same value as that on the surface.
Physics 201 – p. 10/2
Electric Potential and Energy
Charged conductor of arbitrary shape
Charged conducting sphere: 1) V = 4πkσR
on the surface and inside; 2) E = 4πkσ on the
surface and perpendicular to it and zero
inside.
Physics 201 – p. 11/2
Electric Potential and Energy
Charged conductor of arbitrary shape
Charged conducting sphere: 1) V = 4πkσR
on the surface and inside; 2) E = 4πkσ on the
surface and perpendicular to it and zero
inside.
Two spheres with different radii, R1 and R2
and same potential ⇒ σ1 R1 = σ2 R2 ⇒
σ1 = σ2 R2 /R1
Physics 201 – p. 11/2
Electric Potential and Energy
Charged conductor of arbitrary shape
Charged conducting sphere: 1) V = 4πkσR
on the surface and inside; 2) E = 4πkσ on the
surface and perpendicular to it and zero
inside.
Two spheres with different radii, R1 and R2
and same potential ⇒ σ1 R1 = σ2 R2 ⇒
σ1 = σ2 R2 /R1
If R1 ≪ R2 ⇒ σ1 ≫ σ2 ⇒ E1 ≫ E2 ⇒ Sharp
ends of a charged conductor have larger
electric fields.
Physics 201 – p. 11/2
Electric Potential and Energy
Charged conductor of arbitrary shape
Physics 201 – p. 12/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Medical applications
The body is not an ideal conductor ⇒
differences in potential from one place to
another
Physics 201 – p. 13/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Medical applications
The body is not an ideal conductor ⇒
differences in potential from one place to
another
Differences in potential ⇒ Electrocardiograph
and electroencephalograph
Physics 201 – p. 13/2
Electric Potential and Energy
Equipotential surfaces and Electric fields: Moving
from one surface to another
Take two equipotential surfaces which are
very close to each other so that the electric
field is more or less constant:
∆ V = −E∆ s ⇒ E = − ∆∆Vs
Physics 201 – p. 14/2
Electric Potential and Energy
Equipotential surfaces and Electric fields: Moving
from one surface to another
Take two equipotential surfaces which are
very close to each other so that the electric
field is more or less constant:
∆ V = −E∆ s ⇒ E = − ∆∆Vs
∆V
∆s
is a potential gradient. The minus sign
says that the electric field points in the
direction of decreasing potential.
Physics 201 – p. 14/2
Electric Potential and Energy
Equipotential surfaces and Electric fields: Moving
from one surface to another
Take two equipotential surfaces which are
very close to each other so that the electric
field is more or less constant:
∆ V = −E∆ s ⇒ E = − ∆∆Vs
∆V
∆s
is a potential gradient. The minus sign
says that the electric field points in the
direction of decreasing potential.
Change in the electric potential ⇒ electric
field.
Physics 201 – p. 14/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Example
A parallel plate capacitor is composed on two
charged plates, one positive and the other one
negative. A constant electric field points from the
positive plate to the negative plate. Suppose the
potential difference between the 2 plates is 64 V
and that they are separated by 0.032m. By this
we mean ∆V = V− − V+ = −64V . Between the
two plates, one can draw equipotential planes
parallel to the plates. What is the separation
between 2 of such planes if 3.0 V ?
Physics 201 – p. 15/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Solution to Example
Between 2 parallel charged plates, the
electric field is constant. One obtains
−64 V
E = − ∆V
=
−
∆s
0.032m
= 2.0 × 103 V /m
The electric field is pointing in the direction of
decreasing potential.
Physics 201 – p. 16/2
Electric Potential and Energy
Equipotential surfaces and Electric fields:
Solution to Example
Let ∆d be the separation between these two
equipotential surfaces. Since E is constant,
one has
−3.0V
∆d = − ∆V
=
−
E
2.0×103 V /m
= 1.5 × 10−3 m
Physics 201 – p. 17/2
Electric Potential and Energy
Capacitors
Example of a capacitor: two
oppositely-charged parallel plates .
Physics 201 – p. 18/2
Electric Potential and Energy
Capacitors
Example of a capacitor: two
oppositely-charged parallel plates .
Experiment: Increase the charge Q on each
plate ⇒ Direct increase in the potential
difference. ⇒ Q is directly proportional to ∆V .
Physics 201 – p. 18/2
Electric Potential and Energy
Capacitors
The constant of proportionality is called the
capacitance, i.e. the capacity of the device to
store charge.
Q = C|∆V | (6)
C is called the capacitance (a positive
number always).
Unit: 1 f arad(F ) = 1 C/V .
Physics 201 – p. 19/2
Electric Potential and Energy
Capacitors
Numerous applications: For example, a RAM
chip consists of millions of transitor-capacitor
units. Capacitor is charged: 1. Capacitor is
uncharged: 0.
Physics 201 – p. 20/2
Electric Potential and Energy
Capacitors: Calculating capacitances
Parallel-plate capacitor of separation d and
area A:
Three pieces of information:
E = σ/ǫ0 = Q/(ǫ0 A), |∆V | = Ed, and
C = Q/|∆V |.⇒
Physics 201 – p. 21/2
Electric Potential and Energy
Capacitors: Calculating capacitances
Parallel-plate capacitor of separation d and
area A:
Three pieces of information:
E = σ/ǫ0 = Q/(ǫ0 A), |∆V | = Ed, and
C = Q/|∆V |.⇒
Q
Ed
C=
= ǫ0dA
Purely geometrical!
Physics 201 – p. 21/2
Electric Potential and Energy
Capacitors: Example
When a potential difference of 150 V is applied to
the plates of a parallel- plate capacitor, the plates
carry a surface charge density of 30.0 nC/cm2 .
What is the spacing between the plates?
Use C =
ǫ0 A
d .
⇒d=
ǫ0 A
C .
Physics 201 – p. 22/2
Electric Potential and Energy
Capacitors: Example
When a potential difference of 150 V is applied to
the plates of a parallel- plate capacitor, the plates
carry a surface charge density of 30.0 nC/cm2 .
What is the spacing between the plates?
Use C =
ǫ0 A
d .
⇒d=
ǫ0 A
C .
C = Q/|∆V | = σA/|∆V | ⇒
ǫ0 |∆V |
d = σ = 4.42 × 10−6 m
Physics 201 – p. 22/2
Electric Potential and Energy
Capacitors: Dielectric
Electric dipole moments: centers of positive
and negative charges do not coincide. Some
material have permanent electric dipole
moments
Physics 201 – p. 23/2
Electric Potential and Energy
Capacitors: Dielectric
Electric dipole moments: centers of positive
and negative charges do not coincide. Some
material have permanent electric dipole
moments
Insert a slab of such material in between two
oppositely charged plates.
Physics 201 – p. 23/2
Electric Potential and Energy
Capacitors: Dielectric
Between the plates, the negative sides are
attracted to the positive plate and the positive
sides are attracted to the negative plate,
creating an electric field which points in the
opposite direction to that of the external
electric field E0 , and partially cancelling it
inside. ⇒ Inside E, will be smaller than the
electric field without the slab ⇒ Reduction in
the electric field ⇒ Decrease in the voltage ⇒
Increase in the capacitance when the charge
is kept fixed.
Physics 201 – p. 24/2
Electric Potential and Energy
Capacitors: Dielectric
Physics 201 – p. 25/2
Electric Potential and Energy
Capacitors: Capacitance in presence of
Dielectric
Dielectric constant: κ:
κ=
E0
E
Physics 201 – p. 26/2
Electric Potential and Energy
Capacitors: Capacitance in presence of
Dielectric
Dielectric constant: κ:
κ=
E0
E
V0 = E0 d ⇒ V0 = κEd ⇒ V0 = κV .
Physics 201 – p. 26/2
Electric Potential and Energy
Capacitors: Capacitance in presence of
Dielectric
Dielectric constant: κ:
κ=
E0
E
V0 = E0 d ⇒ V0 = κEd ⇒ V0 = κV .
Capacitance in the presence of a dielectric:
C=
Q
V
=
Q
(V0 /κ)
= κC0
Physics 201 – p. 26/2
Electric Potential and Energy
Energy storage
W = 12 QV : Total work done to completely
charge up a capacitor
Physics 201 – p. 27/2
Electric Potential and Energy
Energy storage
W = 12 QV : Total work done to completely
charge up a capacitor
1
2 QV
U=
energy
=
1
2
CV
2
=
Q2
2C :Stored
as potential
Physics 201 – p. 27/2
Electric Potential and Energy
Energy storage
W = 12 QV : Total work done to completely
charge up a capacitor
1
2 QV
U=
energy
=
1
2
CV
2
=
Q2
2C :Stored
as potential
u = U/(Ad) = 21 κǫ0 E 2 : Energy density stored
between the plates
Physics 201 – p. 27/2
Electric Potential and Energy
Some applications
Computer key:
Parallel-plate capacitor filled with a dielectric.
One end is fixed and the other end movable
(attached to a key). Push the key down ⇒
decrease the separation ⇒ increase the
capacitance detected by an electronic circuit
⇒ signal is sent to the computer.
Physics 201 – p. 28/2
Electric Potential and Energy
Some applications
Computer key:
Parallel-plate capacitor filled with a dielectric.
One end is fixed and the other end movable
(attached to a key). Push the key down ⇒
decrease the separation ⇒ increase the
capacitance detected by an electronic circuit
⇒ signal is sent to the computer.
Neurons: Discussed in extraprob1-202.pdf.
Physics 201 – p. 28/2