physics 2112 unit 6: electric potential...example 6.7: ( v for charges) a solid metal conducting...
TRANSCRIPT
Physics 2112
Unit 6: Electric Potential
Today’s Concept:
Electric Potential
(Defined in terms of Path Integral of Electric Field)
Unit 6, Slide 1
Big Idea
Last time we defined the electric potential energy of charge q in an electric field:
∆𝑈𝑎→𝑏 = −න𝑎
𝑏
Ԧ𝐹 ∙ 𝑑Ԧ𝑙 = −න𝑎
𝑏
𝑞𝐸 ∙ 𝑑Ԧ𝑙
The only mention of the particle was through its charge q.
We can obtain a new quantity, the electric potential, which is a PROPERTY OF THE SPACE, as the potential energy per unit charge.
Note the similarity to the definition of another quantity which is also a PROPERTY OF THE SPACE, the electric field.
b
a
baba ldE
q
UV
q
FE
Unit 6, Slide 2
Electricity & Magnetism Lecture 2, Slide 3
We never talk about the “gravitational potential” in 2111. We just talked about a “gravitational force”.
Let’s say we had. What is the magnitude of the gravitational potential close to the surface of the earth?
A) g
B) mgh
C) gh
D) mg
Example 6.2a (Potential from Field)
= 6 N/C|| E
q=+8uC
A +8uC charge is placed in a
downwards pointing electric field
with a magnitude of 6N/C. An
outside force moves the charge
up a distance of 0.4m from point
1 to point 2.
a) What is the force on this charge?
b) How much work was done by the outside force during this
move?
c) How much work was done by the electric field during this
move?
d) What is the change in the electrical potential energy of the
particle?
e) What is the difference in electrical potential between points 1
and 2?
1
2
In the example just worked with a positive
charge q and the field pointing down, we found
that
a) the force on the charge was -4.8 X 10-5Nb) the work done moving the charge up was –1.92 X 10-5Jc) the change in electrical potential energy (EPE) was +1.92 X 10-5Jd) the change in the electrical potential was +2.4V
q
E2
+
+
A. (a)
B. (b)
C. (c)
D. ( d)
E. All of them would change
Which of these values would not change if I redid the
problem, switching q to a negative (-8mC) charge?
1
Question
when we moved the charge upwards 0.4m.
In the example just worked with a positive
charge q and the field pointing down, we found
that
a) the force on the charge was -4.8 X 10-5Nb) the work done moving the charge up was –1.92 X 10-5Jc) the change in electrical potential energy (EPE) was +1.92 X 10-5Jd) the change in the electrical potential was +2.4V
q
E2
+
+
A. (a)
B. (b)
C. (c)
D. ( d)
E. All of them would change
Which of these values would not change if we went
back to the original +8mC charge, but now the field
pointed up?
1
Question
when we moved the charge upwards 0.4m.
In the example just worked with a positive
charge q and the field pointing down, we found
that
a) the force on the charge was -4.8 X 10-5Nb) the work done moving the charge up was –1.92 X 10-5Jc) the change in electrical potential energy (EPE) was +1.92 X 10-5Jd) the change in the electrical potential was +2.4V
q
E2
+
+
What it the electrical potential of the original point 1?
1
Question
when we moved the charge upwards 0.4m.
A. -2.4V
B. 0V
C. 4.8V
D. -4.8V
E. We can’t tell.
Electric Potential from E field
Consider the three points A, B, and C located in a region of constant electric field as shown.
What is the sign of VAC VC VA ?
A) VAC < 0 B) VAC 0 C) VAC > 0
Choose a path (any will do!)
D
Dx
C
D
D
A
CA ldEldEV
00 < xEldEV
C
D
CA
Unit 6, Slide 8
CheckPoint: Zero Electric Field
Electricity & Magnetism Lecture 6, Slide 9
Suppose the electric field is zero in a certain region of space. Which of the following statements best describes the electric potential in this region?
A. The electric potential is zero everywhere in this region.B. The electric potential is zero at least one point in this
region.C. The electric potential is constant everywhere in this
region.D. There is not enough information given to distinguish
which of the above answers is correct.
B
A
BA ldEV
Remember the definition
Example 6.2 (V from point charges)
Unit 6, Slide 10
A B x+Q Q
10cm 10cm 10cm
+5nc -5nc
What is the electrical potential at
points A and B?
Define V = 0 at r =(standard)
Example 6.3 (V from point charges)
Unit 6, Slide 11
C
x+Q Q
10cm 20cm
10cm+5nc -5nc
What is the electrical potential at
point C?
Define V = 0 at r =(standard)
E from V
If we can get the potential by integrating the electric field:
We should be able to get the electric field by differentiating the potential?
b
a
ba ldEV
VE
In Cartesian coordinates:
dx
VEx
dy
VEy
dz
VEz
Electricity & Magnetism Lecture 6, Slide 12
Example 6.4 (E-field above a ring of charge)
Unit 2, Slide 13
a
y
P What is the electrical
potential, V, a distance y
above the center of ring
of uniform charge Q and
radius a? (Assume V = 0
at y = )
y
x
What is the electrical
field, E, at that point?
CheckPoint: Spatial Dependence of Potential 1
Electricity & Magnetism Lecture 6, Slide 14
The electric potential in a certain region is plotted in the following graph
At which point is the magnitude of the E-FIELD greatest?
A. AB. B C. CD. D
CheckPoint: Spatial Dependence of Potential 2
Electricity & Magnetism Lecture 6, Slide 15
The electric potential in a certain region is plotted in the following graph
At which point is the direction of the E-field along the negative x-axis?
A. AB. B C. CD. D
WARNING!!!!
Unit 6, Slide 16
Good news: They’re going to be
really, really simple. Sometimes
so simple we can do them
conceptually.
We’re about to do some
integrals.
Example 6.5: Charged Plates
sR= +6nC/m2sL= -2nC/m2
10cm
30cm
15cm
P W
Two infinite plates are placed
30cm apart and are charged
with the surface charge
densities shown.
What is the electrical potential
difference between point P
which is 10cm to the left of the
negative plate and point W
which is 15cm to the right of
the positive plate?
Example 6.6: (V near line of charge)
Unit 2, Slide 18
An infinitely long solid insulating cylinder of radius a = 4.1 cm
is positioned with its symmetry axis along the z-axis as
shown. The cylinder is uniformly charged with a charge
density ρ = 27.0 μC/m3. Concentric with the cylinder is a
cylindrical conducting shell of inner radius b = 19.9 cm, and
outer radius c = 21.9 cm. The conducting shell has a linear
charge density λ = -0.36μC/m.
What is V(P) – V(R), the potential
difference between points P and R?
Point P is located at
(x,y) = (46.0 cm, 46.0 cm).
Point R is located at
(x,y) = (0 cm, 46.0 cm).
C
Example 6.7: (V for charges)
A solid metal conducting sphere of radius a = 2.5 cm has a net charge Qin = - 3 nC. The sphere is surrounded by a concentric conducting spherical shell of inner radius b = 6 cm and outer radius c = 9 cm. The shell has a net charge Qout = + 2 nC.
What is V0, the electric potential at the center of the metal sphere, given the potential at infinity is zero?
Plan:
Two Step process:
- Step #1: Use Gauss’ Law to calculate E everywhere
- Step #2: Integrate E to get V
cross-section
Electricity & Magnetism Lecture 6, Slide 19
B
A
BA ldEV
o
enc
surface
QAdE
Example 6.6 (V for charges)
A solid metal conducting sphere of radius a = 2.5 cm has a net charge Qin = - 3 nC. The sphere is surrounded by a concentric conducting spherical shell of inner radius b = 6 cm and outer radius c = 9 cm. The shell has a net charge Qout = + 2 nC.
What is V0, the electric potential at the center of the metal sphere, given the potential at infinity is zero?
cross-section
Electricity & Magnetism Lecture 6, Slide 20
E V
Example 6.8: (V for spherical shells)
Point charge q at center of concentric conducting spherical shells of radii a1, a2, a3, and a4. The inner shell is uncharged, but the outer shell carries charge Q.
What is V as a function of r?
Plan:
Two Step process:
- Step #1: Use Gauss’ Law to calculate E everywhere
- Step #2: Integrate E to get V
metal
metal
+Q
a1
a2
a3
a4
+q
cross-section
Electricity & Magnetism Lecture 6, Slide 21
B
A
BA ldEV
o
enc
surface
QAdE
Question
A) 0 B) C)
Electricity & Magnetism Lecture 6, Slide 22
metal
metal
+Q
a1
a2
a3
a4
D) E)
Why?Gauss’ law:
r
0enclosedQ
AdE
+q
cross-section
2
40 )(4
1
ar
Q
2
40 )(2
1
ar
+
2
04
1
r
2
04
1
r
qQ +
0
24
rE+
r > a4 : What is E(r)?
metal
metal
+Q
a1
a2
a3
a4
+q
cross-section
Calculation: Quantitative Analysis
Electricity & Magnetism Lecture 6, Slide 23
a2 < r < a3 :
r
r < a1 : 2
04
1
r
qE
a4 < r :
a3 < r < a4 : 0E
a1 < r < a2 : 0E
2
04
1
r
qQ +
metal
metal
+Q
a1
a2
a3
a4
+q
cross-section
Question
Electricity & Magnetism Lecture 6, Slide 24
r
a3 < r < a4 ?
A)
B)
C)
404
1
a
qQV
+
304
1
a
qQV
+
0V
What is V for
( ) 04 + aV
r
qQV
+
04
1
D)
Example 6.9: (V for Insulator)
A spherical insulator has a total charge +Q and a radius of a.
What is V as a function of r?
insulator
+Q
a
Electricity & Magnetism Lecture 6, Slide 25
Plan:
Two Step process:
- Step #1: Use Gauss’ Law to calculate E everywhere
- Step #2: Integrate E to get V
Electricity & Magnetism Lecture 6, Slide 25
B
A
BA ldEV
o
enc
surface
QAdE
insulator
+Q
a
E
r
Unit 2, Slide 27
Equipotentials
In previous example,
all these points had
same V, same
electrical potential
Line is called “equipotenial”
or “a line of equipotential”.
Topographic Map
Lines on topo map
are lines of “equal
gravitational
potential”
Closer the lines are,
the steeper the hill
Gravity does no
work when you walk
along a brown line.
Equipotential for Point Charge
Equipotentials produced by a point charge
SPACING of the equipotentials indicates STRENGTH of the E field.
Electricity & Magnetism Lecture 6, Slide 29
0V 0elecW 0 ldE
Equipotentials always perpendicular to field lines.
Question
Unit 6, Slide 30
Which of the equipotential
diagrams shown to the left best
describes the spherical insulator
in the described below?
(The colored circle in the center
represents the insulator.)
CheckPoint: Electric Field Lines 1
Electricity & Magnetism Lecture 6, Slide 31
The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines.
At which point in space is the E-field the weakest?
A. AB. BC. CD. D
CheckPoint: Electric Field Lines 3
Electricity & Magnetism Lecture 6, Slide 32
Compare the work done moving a negative charge from A to B and from A to D. Which one requires more work?
The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotentiallines.
A. More work is required to move a negative charge from A to B than from A to D
B. More work is required to move a negative charge from A to D than from A to B
C. The same amount of work is required to move a negative charge from A to B as to move it from A to D
D. Cannot determine without performing the calculation
The diagram to the right shows
equally space equipotential lines 1V
apart. Where on the above diagram
is the electric field the strongest?
A. On the left hand side
B. On the right hand side
C. In the middle
D. It’s constant throughout
E. We can’t tell about the electric field from the equipotential lines.
12V 14V10V
Question
An electric field is represented
by the equipotential surface
lines shown to the right. If a
positive charge is place in this
field, how will it move?
A. Constant velocity to the left
B. Constant velocity to the right
C. Constant acceleration to the left
D. Constant acceleration to the right
E. None of the above
0V-100V
(Each line represent a change of 20V.)
Question
Which of the below drawings of
possible equipotential surfaces
could represent the electrical field
lines shown to the right?
A. (a)
B. (b)
C. (c)
D. (a) or (c) depending on the sign of the test charge
E
(a)(b) (c)
Question
Which of the below drawings of
possible equipotential surfaces
could represent the electrical field
lines shown to the right?
A. (a)
B. (b)
C. (c)
D. (a) or (c)
E. (a) or (b)
E
(a) (b) (c)
10V
50V
50V
10V -10V
-50V
Question