ConcepTest: Electric Potential
ÎWhich
point has the largest potential when Q > 0?
ÎWhich
two points have the same potential?
A and C
‹ (b) B and E
‹ (c) B and D
‹ (d) C and D
‹ (e) no pair
E
Smallest radius
‹ (a)
Same radius
A
C
Q
B
D
E
PHY2049: Chapter 24
23
Multiple Charges: Superposition
Î3
charges: Find total potential at a point in space
+Q
+Q11
− Q3
− Q3
r1
r3
+Q2
x
r2
+Q2
Vtot
‰
−Q3
Q1
Q2
= V1 + V2 + V3 = k + k
+k
r1
r2
r3
V is a scalar
¾ No directions to worry about!
¾ But you do have to watch signs!
PHY2049: Chapter 24
24
ConcepTest: Electric Potential
ÎWhat
is V at point A?
‹ (a)
V>0
‹ (b) V = 0
‹ (c) V < 0
A
B
60 cm
Closer to + charge
40 cm
30 cm
26 cm
Q2 = +50μC
ÎWhat
26 cm
Q1 = −50μC
is V at point B?
‹ (a)
V>0
‹ (b) V = 0
‹ (c) V < 0
Equal distance to both charges
PHY2049: Chapter 24
25
ConcepTest: Electric Potential
ÎAt
which point does V = 0?
‹ (a)
‹ (b)
‹ (c)
‹ (d)
‹ (e)
C
A
D
B
all of the above
All points equidistant
from charges
A
B
C
−Q
+Q
D
PHY2049: Chapter 24
26
ConcepTest: Electric Potential
ÎWhich
configuration gives V = 0 at all points on x axis?
+2μC
+1μC
+2μC
+1μC
x
-1μC
-2μC
A
‹ (a)
+2μC
-2μC
x
-2μC
-1μC
B
x
+1μC
-1μC
C
All points on x axis equidistant
from each pair of charges
A
‹ (b) B
‹ (c) C
‹ (d) All of the above
‹ (e) None of the above
PHY2049: Chapter 24
27
ConcepTest: Fields & Potentials
ÎFind
E and V at the center of the square.
‹ (a)
‹ (b)
‹ (c)
‹ (d)
‹ (e)
E=0
E=0
E≠0
E≠0
E = V
V=0
V≠0
V≠0
V=0
regardless of the value
-Q
+Q
-Q
+Q
PHY2049: Chapter 24
28
ConcepTest: Electric Potential
ÎYou
move a positive charge Q from A to B along the path
shown. What is the sign of the work done by you?
‹ (a)
WAB < 0
‹ (b) WAB = 0
‹ (c) WAB > 0
No change in potential since
distance from center is the same
A
B
PHY2049: Chapter 24
29
Potential of Charge Distribution
ÎGeneralize
superposition to continuous distribution
Vtot
ÎDistribution
‹ Line,
kdq
=∫
r
can be any shape
surface, volume
ÎExpress
dq in terms of charge density
or arc: dq = λds or dq = λdx (λ = linear charge density)
‹ Surface:
dq = σdA
(σ = surface charge density)
‹ Volume:
dq = ρdV
(ρ = volume charge density)
‹ Line
ÎExpress
‹ x,
r in terms of a problem’s “natural” coordinates
θ, r, …
PHY2049: Chapter 24
30
Example: Charged Ring
ÎFind
r
V at a point z above axis of charged ring of radius R
2π k ( λ Rdθ )
kdq
V =∫
=∫
0
r
z 2 + R2
z
V=
R
Q
λ=
2π R
dq = λ Rdθ
r = z 2 + R2
2π k λ R
z 2 + R2
=
kQ
z 2 + R2
∂V
kQz
=
Ez = −
∂z
z 2 + R2
(
For z R E z =
PHY2049: Chapter 24
)
3/ 2
kQ
z2
31
Example: Charged Line
ÎFind
V above midpoint of line of charge Q, length L
P
λ =Q/L
r = x2 + y 2
y
L
L / 2 k ( λ dx )
kdq
=∫
V =∫
−L / 2
r
y 2 + x2
x
⎛
= k λ ln ⎜
⎜
⎝
PHY2049: Chapter 24
dq = λ dx
y 2 + L2 / 4 + L / 2 ⎞
⎟
y 2 + L2 / 4 − L / 2 ⎟⎠
32
Charged Line: Limit of L y
ÎRationalize
⎛
ln ⎜
⎜
⎝
ÎFor
expression inside ln()
⎡
⎢
y 2 + L2 / 4 + L / 2 ⎞
⎟ = ln ⎢
⎢
y 2 + L2 / 4 − L / 2 ⎟⎠
⎢⎣
(
)
2⎤
y + L /4 + L/2 ⎥
⎥
2
y
⎥
⎥⎦
2
2
Ly
⎛ L2 ⎞
→ ln ⎜ 2 ⎟
⎜y ⎟
⎝ ⎠
⎛L⎞
V = 2k λ ln ⎜ ⎟
⎝ y⎠
PHY2049: Chapter 24
33
Charged Line (cont)
ÎCalculate
y component of electric field at midpoint
⎛ y 2 + L2 / 4 + L / 2 ⎞
⎟
V = k λ ln ⎜
⎜ y 2 + L2 / 4 − L / 2 ⎟
⎝
⎠
∂V
kλ L
Ey = −
=
∂y y y 2 + L2 / 4
PHY2049: Chapter 24
Agrees with calculation
in previous chapter
34
Example: Charged Disk
ÎFind
V at a point z above axis of charged disk of radius R
z
r
ρ
σ=
R 2π k (σρ d ρ dθ )
kdq
=∫ ∫
V =∫
0 0
r
z2 + ρ 2
Q
πR
dq = σ dA
2
R
R kσ
( 2πρ d ρ )
0
z2 + ρ 2
V =∫
(surface charge density)
= σ ( ρ d ρ dθ )
V = 2π kσ
PHY2049: Chapter 24
(
z 2 + R2 − z
)
35
Charged Disk (cont)
ÎAnother
approach: treat disk as
concentric charged rings
ρ
( 2πρ d ρ )
0
z2 + ρ 2
V =∫
z
r
R kσ
R
V = 2π kσ
dq = σ dA
(
z 2 + R2 − z
)
= σ ( 2πρ d ρ )
Follows from
A = πρ 2 ⇒ dA = 2πρ d ρ
PHY2049: Chapter 24
36
Charged Disk (cont)
ÎCalculate
z component of electric field
V = 2π kσ
(
z 2 + R2 − z
)
⎛
z
∂V
Ez = −
= 2π kσ ⎜1 −
⎜
2
2
∂z
z +R
⎝
ÎWhen
⎞
⎟⎟
⎠
z very small
σ
Ez 2π kσ =
2ε 0
Just like sheet of charge
PHY2049: Chapter 24
37
V
Dipole
−Q
x
+Q
kQ kQ
V =−
+
r1
r2
No equilibrium since E is never 0
Where are equilibrium points?
V
kQ kQ
V=
+
r1
r2
Like charges
Equilibrium is at x = 0, since E = 0
E is – dV/dx
+Q
+Q
x
PHY2049: Chapter 24
38
Conductors are Equipotentials
ÎNo
work to move along conductor
‹W
ÎBut
= 0 = −qΔVAB ⇒ V is constant in conductor
E = 0 inside surface bounded by conductor
‹ VC
= VA ⇒ V is constant within enclosed volume
A
B
C
PHY2049: Chapter 24
39
Conductors in Electrostatic Equilibrium
ÎElectric
‹ if
field is zero everywhere inside the conductor
E ≠ 0, then charges would move – no equilibrium!!
ÎExcess
charge on isolated conductor is only on surface
‹ Mutual
ÎElectric
‹ If
repulsion pushes the charges apart
field is perpendicular to the surface of a conductor
a parallel component existed, charges would move!!
ÎFor
irregular shaped conductors, charge density is highest
near sharp points, i.e. the field strength is greater there
PHY2049: Chapter 24
40
Spherical Shell
+
-
+
-
-
+ -
+
- +
+Q
+ -
+
-
+
-
+
+
Inner radius = r1
Outer radius = r2
¾ What is charge on inner shell?
−Q
¾ What is charge on outer shell?
+Q
¾ What is V vs radius?
Constant from 0 < r < r2
Falls as kQ / r for r > r2
PHY2049: Chapter 24
41
ConcepTest: Electric Energy
ÎA
positively charged rod is held near a neutral conducting
sphere. A positively charged particle is moved from A to B
(A, B both on sphere).
ÎThe
mechanical work required to cause this motion is
‹ (a)
positive
‹ (b) zero
All points on sphere are at same potential
‹ (c) negative
‹ (d) depends on the path taken from A to B
‹ (e) cannot be determined without more information
PHY2049: Chapter 24
42
ConcepTest: Electric Energy
ÎA
positively charged rod is held near a neutral conducting
sphere. A positively charged particle is moved from A to B
(A is on sphere).
ÎThe
mechanical work required to cause this motion is
‹ (a)
positive
Must push against electrostatic force
‹ (b) zero
‹ (c) negative
‹ (d) depends on the path taken from A to B
‹ (e) cannot be determined without more information
PHY2049: Chapter 24
43
ConcepTest: Electric Energy
ÎA
positively charged rod is held near a neutral conducting
sphere. A positively charged particle is moved from A to B
(A on sphere).
ÎThe
electrostatic work done on the particle is
‹ (a)
positive
‹ (b) zero
‹ (c) negative Electrostatic force is against direction of motion
‹ (d) depends on the path taken from A to B
‹ (e) cannot be determined without more information
PHY2049: Chapter 24
44
ConcepTest: Electric Potential
ÎA
positively charged rod is held near a neutral conducting
sphere (A on sphere).
ÎThe
potential change from A to B is:
‹ (a)
positive
Higher potential near + charge
‹ (b) zero
‹ (c) negative
‹ (d) depends on the path taken from A to B
‹ (e) cannot be determined without more information
PHY2049: Chapter 24
45
ConcepTest: Electrostatics
ÎTwo
charged metal spheres are connected by a copper
wire. Note that rA > rB.
ÎWhich
quantity must be the same for both spheres?
‹ (a)
potential at the surface
‹ (b) charge on the sphere
‹ (c) surface charge density
‹ (d) field at the surface
‹ (e) more than one of the above.
PHY2049: Chapter 24
46
ConcepTest: Electrostatics
ÎTwo
charged metal spheres are connected by a copper
wire. Note that rA > rB.
ÎCompare
qA to qB
qA > qB
‹ (b) qA < qB
‹ (c) qA = qB
‹ (a)
‹ (d)
Potential is same, so kqA/rA = kqB/rB
Need more information
PHY2049: Chapter 24
47
ConcepTest: Electric Potential
ÎA
solid spherical conductor is given a net nonzero charge.
The electric potential of the conductor is
‹ (a)
largest at the center.
‹ (b) largest on the surface.
‹ (c) largest somewhere between center and surface.
‹ (d) constant throughout the volume.
PHY2049: Chapter 24
48
Review: Electric Potential
+Q
-4Q
What charge will make the
potential zero at X ?
x
+4Q
Charge = ??
Charge = ??
What charge will make the
potential zero at X?
Charge = ??
-2Q
x
What charge will make the
potential zero at X?
-Q
+5Q
r
PHY2049: Chapter 24
2r
x
+3Q
r
49
PHY2049: Chapter 24
50