PHY2049: Chapter 24 23
ConcepTest: Electric Potential Which point has the largest potential when Q > 0?
Which two points have the same potential?(a) A and C(b) B and E(c) B and D(d) C and D(e) no pair
AC
BD
E
Q
E
Smallest radius
Same radius
PHY2049: Chapter 24 24
Multiple Charges: Superposition3 charges: Find total potential at a point in space
− Q3
+Q1
+Q2
xr1
r3r2
31 21 2 3
1 2 3tot
QQ QV V V V k k kr r r
−= + + = + +
V is a scalarNo directions to worry about!But you do have to watch signs!
+Q1
+Q2
−Q3
PHY2049: Chapter 24 25
ConcepTest: Electric Potential What is V at point A?
(a) V > 0 (b) V = 0 (c) V < 0
What is V at point B?(a) V > 0(b) V = 0(c) V < 0
Closer to + charge
Equal distance to both charges
A B
Q2 = +50μC Q1 = −50μC
30 cm
26 cm 26 cm
40 cm
60 cm
PHY2049: Chapter 24 26
ConcepTest: Electric Potential At which point does V = 0?
(a) C (b) A (c) D (d) B (e) all of the above
A
C
B
D
+Q −QAll points equidistantfrom charges
PHY2049: Chapter 24 27
ConcepTest: Electric Potential Which configuration gives V = 0 at all points on x axis?
(a) A(b) B(c) C(d) All of the above (e) None of the above
A
x
+2μC
-2μC
+1μC
-1μCB
x
+2μC
-1μC
+1μC
-2μCC
x
+2μC
-1μC
-2μC
+1μC
All points on x axis equidistantfrom each pair of charges
PHY2049: Chapter 24 28
ConcepTest: Fields & Potentials Find E and V at the center of the square.
(a) E = 0 V = 0(b) E = 0 V ≠ 0(c) E ≠ 0 V ≠ 0(d) E ≠ 0 V = 0(e) E = V regardless of the value
-Q
-Q +Q
+Q
PHY2049: Chapter 24 29
ConcepTest: Electric Potential You 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
A B
No change in potential sincedistance from center is the same
PHY2049: Chapter 24 30
Potential of Charge DistributionGeneralize superposition to continuous distribution
Distribution can be any shapeLine, surface, volume
Express dq in terms of charge densityLine 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)
Express r in terms of a problem’s “natural” coordinatesx, θ, r, …
totkdqVr
= ∫
PHY2049: Chapter 24 31
Example: Charged RingFind V at a point z above axis of charged ring of radius R
( )2
0 2 2
k RdkdqVr z R
π λ θ= =
+∫ ∫
2Q
Rλ
π=
2 2 2 2
2 k R kQVz R z R
π λ= =
+ +
z
( )3/ 22 2z
V kQzEz z R
∂= − =
∂ +
2 2
dq Rd
r z R
λ θ=
= +
R
r
2For zkQz R Ez
=
PHY2049: Chapter 24 32
Example: Charged LineFind V above midpoint of line of charge Q, length L
( )/ 2
/ 2 2 2
L
L
k dxkdqVr y x
λ−
= =+
∫ ∫
/Q Lλ =
2 2
2 2
/ 4 / 2ln
/ 4 / 2
y L Lk
y L Lλ
⎛ ⎞+ +⎜ ⎟=⎜ ⎟+ −⎝ ⎠
y
P
xL dq dxλ=
2 2r x y= +
PHY2049: Chapter 24 33
Charged Line: Limit of L y
Rationalize expression inside ln()
For L y2
2ln Ly
⎛ ⎞→ ⎜ ⎟⎜ ⎟
⎝ ⎠
( )22 22 2
22 2
/ 4 / 2/ 4 / 2ln ln
/ 4 / 2
y L Ly L Lyy L L
⎡ ⎤+ +⎛ ⎞ ⎢ ⎥+ +⎜ ⎟ = ⎢ ⎥
⎜ ⎟ ⎢ ⎥+ −⎝ ⎠ ⎢ ⎥⎣ ⎦
2 ln LV ky
λ⎛ ⎞
= ⎜ ⎟⎝ ⎠
PHY2049: Chapter 24 34
Charged Line (cont)Calculate y component of electric field at midpoint
2 2
2 2
/ 4 / 2ln
/ 4 / 2
y L LV k
y L Lλ
⎛ ⎞+ +⎜ ⎟=⎜ ⎟+ −⎝ ⎠
2 2 / 4y
V k LEy y y L
λ∂= − =
∂ +Agrees with calculationin previous chapter
PHY2049: Chapter 24 35
Example: Charged DiskFind V at a point z above axis of charged disk of radius R
( )2
0 0 2 2
R k d dkdqVr z
π σρ ρ θ
ρ= =
+∫ ∫ ∫
( )0 2 2
2R k dV
z
σ πρ ρ
ρ=
+∫
( )2 22V k z R zπ σ= + −
2 (surface charge density)QR
σπ
=
z
( )dq dA
d dσσ ρ ρ θ
=
=
R
r
ρ
PHY2049: Chapter 24 36
Charged Disk (cont)Another approach: treat disk as
concentric charged rings
z
R
r
ρ
( )2dq dA
dσσ πρ ρ
=
=
( )0 2 2
2R k dV
z
σ πρ ρ
ρ=
+∫
( )2 22V k z R zπ σ= + −
Follows from 2 2A dA dπρ πρ ρ= ⇒ =
PHY2049: Chapter 24 37
Charged Disk (cont)Calculate z component of electric field
When z very small
( )2 22V k z R zπ σ= + −
2 22 1z
V zE kz z R
π σ⎛ ⎞∂
= − = −⎜ ⎟⎜ ⎟∂ +⎝ ⎠
02
2zE k σπ σε
= Just like sheet of charge
PHY2049: Chapter 24 38
Dipole
Like charges
Where are equilibrium points?
V
x
−Q
+Q
V
x+Q+Q
1 2
kQ kQVr r
= − +
1 2
kQ kQVr r
= +
No equilibrium since E is never 0
Equilibrium is at x = 0, since E = 0E is – dV/dx
PHY2049: Chapter 24 39
Conductors are EquipotentialsNo work to move along conductor
W = 0 = −qΔVAB ⇒ V is constant in conductor
But E = 0 inside surface bounded by conductorVC = VA ⇒ V is constant within enclosed volume
A
C
B
PHY2049: Chapter 24 40
Conductors in Electrostatic EquilibriumElectric field is zero everywhere inside the conductor
if E ≠ 0, then charges would move – no equilibrium!!
Excess charge on isolated conductor is only on surfaceMutual repulsion pushes the charges apart
Electric field is perpendicular to the surface of a conductorIf a parallel component existed, charges would move!!
For irregular shaped conductors, charge density is highest near sharp points, i.e. the field strength is greater there
PHY2049: Chapter 24 41
Spherical Shell
+Q
-
-
--
-
--
-- -
++
+
++
++
+
+
+ What is charge on inner shell?−Q
What is charge on outer shell?+Q
What is V vs radius?Constant from 0 < r < r2Falls as kQ / r for r > r2Inner radius = r1
Outer radius = r2
PHY2049: Chapter 24 42
A 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).
The mechanical work required to cause this motion is (a) positive(b) zero(c) negative(d) depends on the path taken from A to B(e) cannot be determined without more information
ConcepTest: Electric Energy
All points on sphere are at same potential
PHY2049: Chapter 24 43
ConcepTest: Electric EnergyA positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A is on sphere).
The mechanical work required to cause this motion is (a) positive(b) zero(c) negative(d) depends on the path taken from A to B(e) cannot be determined without more information
Must push against electrostatic force
PHY2049: Chapter 24 44
A positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B(A on sphere).
The electrostatic work done on the particle is(a) positive(b) zero(c) negative(d) depends on the path taken from A to B(e) cannot be determined without more information
ConcepTest: Electric Energy
Electrostatic force is against direction of motion
PHY2049: Chapter 24 45
A positively charged rod is held near a neutral conducting sphere (A on sphere).
The potential change from A to B is:(a) positive(b) zero(c) negative(d) depends on the path taken from A to B(e) cannot be determined without more information
ConcepTest: Electric Potential
Higher potential near + charge
PHY2049: Chapter 24 46
Two charged metal spheres are connected by a copper wire. Note that rA > rB.
Which 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.
ConcepTest: Electrostatics
PHY2049: Chapter 24 47
Two charged metal spheres are connected by a copper wire. Note that rA > rB.
Compare qA to qB(a) qA > qB
(b) qA < qB
(c) qA = qB
(d) Need more information
ConcepTest: Electrostatics
Potential is same, so kqA/rA = kqB/rB
PHY2049: Chapter 24 48
A 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.
ConcepTest: Electric Potential
PHY2049: Chapter 24 49
Review: Electric Potential
What charge will make the potential zero at X ?
What charge will make thepotential zero at X?
What charge will make thepotential zero at X?
-2Q +5Q
Charge = ??
x
+4Q
-4Q +Q
Charge = ??
x
+3Q-Q
r r
2r
Charge = ??
x
PHY2049: Chapter 24 50
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