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Electricity and magnetism
Lecture 3. Electric Potential and Energy
Outline
• Understand the conservative force,
• Relationship between work, potential energy, and potential,
• Calculate potential of point charges and other symmetric charge sources
Gauss’ Law states that the electric flux through any closed surface is equal to the net charge inside the surface divided by εo
2
q2
q3
1
Review
Gauss’s Law can be used to calculate electric fields of highly symmetric
distributions of charge.
Work Energy Relationship
• Work done by a constant force:
• Work done by a conservative force is equal to the negative of the change in potential energy
• Both electrostatic force and gravitational force are conservative force
)()(
total
iiffnc
cnc
PEKEPEKEPEKEW
KEWWW
xFW
)(constant ifc xxFPEW
xdFW
• Energy conservation law (general):
3
or
i: initial
f: final
Change in Kinetic energy
Change in total energy
Compare electric potential energy and gravitational potential energy
– The behavior of a point charge in a uniform electric field is analogous to the motion of a baseball in a uniform gravitational field.
4
Change in potential energy: Definition
• Let’s first consider a uniform
field between the two plates
• As the charge moves from A to
B, work is done on it
• Work done by electric force
W = F ∆x = q Ex (xf – xi)
• Change in potential energy
ΔPE = – W = – q Ex x
This is only for a uniform field. In more general case, we do the following for the work done by electric force
f
i
x
xxdxqEW
5
Charge Movements and the Energy Change
• When the electric field is directed downward, point B is at a lower potential than point A
• A positive test charge that moves from A to B loses electric potential energy. It will gain the same amount of kinetic energy as it loses in potential energy.
• If a negative charge moves from point A to B, it gains potential energy, so positive external work W must be done to overcome the .
For negative charge
0W
)(
ext
ext
EdqKE
WEdqKEPEKE
• Electric potential energy is characteristic of the charge-field system – Due to an interaction between the field and the charge placed in the field
6
qEdKEW
WqEdKEPEKE
ext
ext
For positive charge
can be 0
Charge Movements and the Energy Change
• When a positive charge is in an electric field, – It spontaneously moves along the direction of the field, namely, from higher potential to
lower potential, and its electrical potential energy decreases (case in Figure a below)
– If it moves opposite to the electric field direction, namely, from lower potential to higher potential, its electrical potential energy increases (case in Figure b below). To make this movement, external work is needed
7 Question: What happens to the kinetic energy if the charge moves from rest?
Charge Movements and the Energy Change
8
• When a negative charge is in an electric field, – It spontaneously moves opposite to the direction of the field, namely, from lower
potential to higher potential, and its electrical potential energy decreases (case in Figure b below)
– If it moves along the electric field direction, namely, from higher potential to lower potential, its electrical potential energy increases (case in Figure a below). To make this movement, external work W is needed
Quick quiz If a negatively charged particle is placed at point B and given a very small kick to the right, what will its subsequent motion be?
1 2 3 4
0% 0%0%0%
1. It will go to the right and never return
2. It will go to the left
3. It will remain at point B
4. It will oscillate around point B
What if it is a positively charged particle?
Countdown
20
Electric Potential Energy is Path Independent
Because electric force is conservative force, the electric potential difference is path independent
The electric potential is the same whether q0 moves in a radial line (left figure) or along an arbitrary path (right figure).
10
Definition of Electric Potential
• The potential difference between points A and B is defined as the change in the potential energy (final value minus initial value) of a charge q moved from A to B divided by the size of the charge
• Potential difference is not the same as potential energy. They have different units!
• The electric potential is independent of the path between points A and B
• When we talk about potential, it is actually always “relative”!
• It is customary to choose a reference potential: at r = ∞ , V = 0 and PE =0. Then the potential at some point r away from a point source charge q can be reformulated as
q
PEVVV
11
(Unit: J/C or V) q
PEVVV AB
• A potential (and field) exists at some point in space, no matte whether there is a test charge
• Both electric potential energy and potential difference are scalar quantities
Finding Electric Potential from Electric Field
12
b
aBA ldEVV
For a point charge q, the electric field is
rr
qkE
2
e
set V = 0, so the potential at any point
with distance r is:
r
qkrd
r
qkrdEVV
rr
e
2
e
General formula:
Electric potential can be calculated from electric field by integration.
Question: What about sign of V?
For a uniform-charged conducting sphere
Finding Electric Potential from Electric Field
• Outside the sphere, the electric field and potential are similar to the case of point charge
• Electric field inside the sphere is zero, so potential difference is zero constant potential from inside up to the surface (we will see this again later)
13
Electric Field and Electric Potential Depend on Distance
• For a point source charge q = 1.11 x 10-10 C
• The electric field is proportional to 1/r2
• The electric potential is proportional to 1/r
• Both are intrinsic property of the source charge q, independent of whether or not there is test charge
14
Relationship Chart (for point charge)
Electric charge Electric charge
x displacement
x displacement
2
1
r
qqkF ee
r
qqkPE e
1
2
1
r
qk
q
FE e
e
r
qk
q
PEV e
1
15
Electrical Potential Energy (Two point charges)
• V1 is the electric potential due to q1 at some point P
• The work required to bring q2 from infinity to P without acceleration is q2V1
• This work is equal to the potential energy of the two particle system
r
qqkVqPE e
2112
16
• If the charges have the same sign, PE is positive – Positive external work must be done to force the two charges near one
another
• If the charges have opposite signs, PE is negative – Negative external work must be done to hold back the unlike charges
from accelerating as they are brought close together
Electrical potential energy (multiple point charges)
• The total electric potential at some point a due to several point charges is the algebraic sum of the electric potentials due to the individual charges
– The algebraic sum is used because potentials are scalar quantities
– The potential at point a is associated to the energy cost by moving q0 from infinity to point a (and then divided by q0)
3
3
2
2
1
1
0
0
r
q
r
q
r
qk
q
PEV e
q
Question: What is the total kinetic energy of the 3 charges after a long time if they are released and free to move?
17
23
32
13
31
12
21total
r
r
r
qqkPE e
• But this is NOT the total electric potential of the original three charges. The energy cost by moving the three charges together from infinity is
Quick quiz
q1 = 5.00 µC q2 = -2.00 µC
If the electric potential is taken to be zero at infinity, find the total electric potential due to these chares at point P. How much work is required to bring a third point charge of 4.00 µC from infinity to P?
V 106.7V
V 10360.0
V 1012.1
3
21tot
4
2
22
4
1
11
VVV
r
qkV
r
qkV
e
e
J 100.3
)(
2
3
33
P
P
Vq
VVqVqPEW
18
Quick quiz Suppose three protons lie on the x-axis, at rest relative to one another at a given instant of time. If proton q3 on the right is released while the others are held fixed in place, find a symbolic expression for the proton’s speed at infinity.
0
2
0
2
12
21
0
2
23
32
13
31
12
21
2
3
2
5
r
qkPE
r
qk
r
qqkPE
r
qk
r
qqk
r
qqk
r
qqkPE
e
eef
eeeei
0
2
0
22
3
2
3
2
1
0
mr
ekv
r
qkmv
PEKE
e
e
19 0
2
0
2
23
32
13
31
2
3
0
2
3
r
qkPE
PE
r
qk
r
qqk
r
qqkPE
e
f
eeei
Alternatively, we can take q3 as a test charge,
The Electron Volt
• It is an energy unit.
• The electron volt (eV) is defined as the energy that an electron gains when accelerated through a potential difference of 1 V
• 1 eV = 1.6 x 10-19 C · (J/C) = 1.6 x 10-19 J
– Electrons in normal atoms have energies of 10’s of eV
– Excited electrons have energies of 1000’s of eV
– High energy gamma rays have energies of millions of eV
20
Equipotential Surfaces
• Since W = -q(VB – VA), net work is zero when moving a charge between two points that are at the same electric potential, namely, W = 0 when VA = VB
• A charge can be moved between any two points at the same electric potential without doing any work
21
• An equipotential surface is a surface on which all points are at the same potential – No work is required to move a charge at a constant speed on an
equipotential surface
– The electric field at every point on an equipotential surface is perpendicular to the surface
Equipotential surfaces and field lines
• Equipotential surfaces can be schematic drawn together with field lines
• Electric field lines and equipotential surfaces are always mutually perpendicular.
22
Question: What happens if a charged particle moves along the equipotential surface?
Conductors in Equilibrium
• The conductor has an excess of positive charge
• All of the charge resides at the surface • E = 0 inside the conductor • The electric field just outside the conductor
is perpendicular to the surface • All points on the surface of a charged
conductor in electrostatic equilibrium are at the same potential
• The potential everywhere inside the conductor is constant and equal to its value at the surface
• The whole conductor in equilibrium is equipotential
090cos B
A
B
A
B
AAB dsqEsdEqsdFW
23
Equipotentials and conductors
• When all charges are at rest: the surface of a conductor is always an equipotential surface.
the electric field just outside a conductor is always perpendicular to the surface (see figures below).
the entire solid volume of a conductor is at the same potential.
24
If the electric field just outside a conductor had a tangential component E||, a charge could move in a loop with net work done.
A Cavity within a Conductor
Because VB – VA = 0, the integral of E.ds must be zero for all paths
between any two points A and B on the conductor. A cavity surrounded
by conducting walls is a field free region as long as no charges are inside
the cavity.
B
B AA
V V d E s
Consider a conductor of arbitrary shape
as shown. The electric field inside the
cavity must be zero regardless of the
charge distribution on the outside
surface of the conductor.
Potential gradient
26
We can also write the vector form of the definition for two points in 3-D space
𝑉𝑏 − 𝑉𝑎 = − 𝐸. 𝑑𝑠 =
𝑏
𝑎
− (𝐸𝑥𝑑𝑥 + 𝐸𝑦𝑑𝑦 + 𝐸𝑧𝑑𝑧)
𝑏
𝑎
Conversely, given the potential V(x, y, z), we can obtain the components of the electric field
𝐸𝑥 = −𝜕𝑉
𝜕𝑥 ; 𝐸𝑦 = −
𝜕𝑉
𝜕𝑦 ; 𝐸𝑧 = −
𝜕𝑉
𝜕𝑧
Here, 𝜕𝑉
𝜕𝑥 denotes a partial derivative – differentiating w.r.t. x and treating variables y
and z in V as constants. It applies similarly to 𝜕𝑉
𝜕𝑦 and
𝜕𝑉
𝜕𝑧. If the electric field only
varies in the radial direction with r, then
𝐸𝑟 = −𝜕𝑉
𝜕𝑟.
The negative sign is associated with the fact that the electric field is always pointing from higher potential to lower potential. We note that another unit for the electric field is volts per metre (V/m).
(a) Find an expression for the electric potential
at point P located on the perpendicular
central axis of a uniformly charged ring of
radius a and total charge Q.
(b) Find an expression for the magnitude of the
electric field at point P.
Example: Electric Potential Due to a Uniformly Charged Ring
(a) Charge element dq at a distance from point P. 2 2x a
2 2 2 2
2 2
dq dq kV k k dq
r x a x a
kQV
x a
Since each element dq is at the same distance from P
(b) Use Ex = dV/dx
12 2 2
32 2 2
32 2 2
1 22
x
x
dV dE kQ x a
dx dx
kQ x a x
kQxE
x a
• What about Ey and Ez ?
• What is the electric potential at the center of the ring?
• What is the electric field at the center of the ring?
Question:
Summary
Potential Difference and Electric Potential - The change in the electric potential energy of a system is given by (1D): - The difference in electric potential between two points A and B is where PE is the change in electrical potential energy as a charge q moves between A and B. The units of potential difference are joules per coulomb, or volts; 1 J/C = 1 V.
29
xqEWPE xAB
q
PEVVV
AB
Two ways to find electric potential:
b
aBA ldEVV
• Sum up potential from continuous point charges (see next slide):
• From electric field, then times displacement:
r
dqkV e
Summary
Electric Potential and Potential Energy by Point Charges - The electric potential of a point charge q at distance r is
If more than one charges, then sum up algebraically. – this gives another way to find out potential
- The electric potential energy of a pair of point charges separated by distance r is:
30
r
qkV e
r
qqkPE e
21
Equipotential Surfaces Any surface on which the potential is the same at every point is called an equipotential surface. The electric field is always oriented perpendicular to an equipotential surface.
