POTENTIAL
Class Activities: Potential (slide 1)
Class Activities: Potential (slide 2)
Today: Voltage or “Electric Potential”
The 1120 version:
• Voltage V = kq/r from a point charge
• Voltage = potential energy/charge
• DV is “path independent”
• E = -dV/dr
Potential and E-field
V(r) º -
r
ò
E(r ') · dr '
origin
r
ò
qV (r) º -
origin
r
=-
ò
qE(r ') · dr '
Fon q · dr '
origin
= external work done on charge to move it
Potential and E-field
V(r) º -
r
ò
E(r ') · dr '
origin
E(r) = -ÑV(r)
The fact that V is well defined arises from
ò E(r)·dl = 0
Which is exactly the same as (!)
Ñ ´ E(r) = 0
Potential and E-field
r
ò
V(r) = -
E(r ') · dr '
origin
E(r) = -ÑV(r)
One way to find V(r) (given charges):
V(r) =
1
4pe0
òòò
r (r')
Â
dt '
One way to find V(r) (given charges):
V(r) =
1
4pe0
V(r) =
òòò
1
4pe 0
r (r')
Â
òòò
dt '
dq
Â
Voltage V = kq/r from a point charge !
So, what does
V(r) º -
r
ò
E(r ') · dr '
origin
Have to do with
ò E(r)·dl = 0
And why is that the same as
Ñ ´ E(r) = 0
Fundamental theorem:
æ df ö
ò a çè dx ÷ø dx = f (b) - f (a)
b
Divergence theorem:
òòò (Ñ· F)dt = òò F · dA
V
S
Stoke’s (or “curl”) theorem:
òò (Ñ ´ F) · dA = ò F · dl
S
Common vortex
Feline Vortex
1.8
What is the curl of this vector field, V
in the region shown below?
V = c ĵ
A. non-zero everywhere
B. Zero at some points,
non-zero others
C. zero curl everywhere
shown
1.8
What is the curl of this vector field, V
in the region shown below?
V = c ĵ
A. non-zero everywhere
B. Zero at some points,
non-zero others
C. zero curl everywhere
shown
1.8
What is the curl of this vector field, V
in the region shown below? V = c ĵ
é 1 ¶vz ¶vf ù
Ñ´V =ê
ú ŝ
ë s ¶f ¶z û
é ¶vs ¶vz ù ˆ 1 é ¶(svf ) ¶vs ù
+ê
ú ẑ
úf + ê
ë ¶z ¶s û
s ë ¶s
¶f û
Why is
Ñ ´ E(r) = 0
1.8
E(r) =
q
4pe0 r
2
r̂
1 é¶
¶vq ù
Ñ´V =
ê (sin q vf ) ú r̂
r sinq ë ¶q
¶f û
1 é 1 ¶vr ¶(rvf ) ù ˆ 1 é ¶(rvq ) ¶vr ù
+ ê
ĵ
úq + ê
ú
r ë sin q ¶f
¶r û r ë ¶r
¶q û
Ñ´E =
0!
2.43b/1.7b
Which of the following could be a static physical Efield in a small region?
I
II
A) Only I B) Only II
C) Both
D) Neither
E) Cannot answer without further info
E(r) =
1
4pe 0
òòò
r(r ') Â̂
Â
2
 = r - r' = (x - x', y - y', z - z')
dt '
E(r) =
1
4pe 0
òòò
r(r ') Â̂
Â
2
dt '
(with  = r - r' = (x - x', y - y', z - z'))
It is also true that
Â̂
1
= -Ñ
2
Â
Â
æ ¶ ¶ ¶ö
where Ñ = çè ¶x , ¶y , ¶z ÷ø
E(r) =
1
4pe 0
òòò
r(r ') Â̂
Â
2
dt '
(with  = r - r' = (x - x', y - y', z - z'))
It is also true that
Â̂
1
= -Ñ
2
Â
Â
æ ¶ ¶ ¶ö
where Ñ = çè ¶x , ¶y , ¶z ÷ø
æ
1 ö
E(r) =
r (r')çç -Ñ ÷÷ dt '
òòò
4pe0
Âø
è
1
E(r) =
1
4pe 0
òòò
r(r ') Â̂
Â
2
dt '
(with  = r - r' = (x - x', y - y', z - z'))
It is also true that
Â̂
1
= -Ñ
2
Â
Â
æ ¶ ¶ ¶ö
where Ñ = çè ¶x , ¶y , ¶z ÷ø
Question: is the following mathematically ok?
æ
1
1 ö
1
r(r ')
E(r) =
r(r ') ç -Ñ ÷ dt ' = -Ñ
dt '
òòò
òòò
4pe 0
Âø
4pe 0
Â
è
A) Yes
B) No
C) ???
A uniformly charged ring, in the xy plane,
centered on the origin, has radius a and total
charge Q. V(r =) = 0.
What is the voltage at z on the z-axis?
z
kQ
A)
a
kQ
C)
a 2 + z2
kQ
B)
z
a
E) None of these
kQ
D) 2 2
a +z
A uniformly charged ring, in the xy plane,
centered on the origin, has radius a and total
charge Q. V(r =) = 0.
What is the voltage at z on the z-axis?
z
kQ
A)
a
kQ
C)
a 2 + z2
Â
kQ
B)
z
a
dq
E) None of these
kQ
D) 2 2
a +z
Summary:
Def of potential:
r
V(r) º - ò E(r ') · dr '
r0
How do you compute it:
What good is it?
V(r) =
1
4pe0
òòò
E(r) = -ÑV(r )
Where did it come from? Ñ ´ E(r) = 0
Which by Stoke’s theorem is
mathematically equivalent to:
ò
any loop
E(r) · dr = 0
r (r')
Â
dt '
Tutorial
When you are DONE, click in:
What is the slope of V(r) at the origin?
A)
B)
C)
D)
Positive
Negative
Zero
It depends!
Be prepared to explain/defend your answer!
2.26
A spherical shell has a uniform positive charge
density on its surface. (There are no other
charges around)
What is the electric field inside the sphere?
A: E=0 everywhere inside
B: E is non-zero everywhere in the
sphere
C: E=0 only at the very center, but
non-zero elsewhere inside the sphere.
D: Not enough info given
2.44
E(r) = -ÑV(r )
r
ò
V(r) º -
E(r ') · dr '
origin
Could this be a plotr of |E|(r)? Or V(r)?
(for SOME physical situation?)
A) Could be E(r), or V(r)
B) Could be E(r), but can't be V(r)
C) Can't be E(r), could be V(r)
D) Can't be either
E) ???
The voltage is zero at a point in space.
You can conclude that :
A) The E-field is zero at that point.
B) The E-field is non-zero at that point
C) You can conclude nothing at all about the Efield at that point
The voltage is constant everywhere
along a line in space.
V=constant
You can conclude that :
A) The E-field has constant magnitude along that
line.
B) The E-field is zero along that line.
C) You can conclude nothing at all about the
magnitude of E along that line.
y
Consider an infinitesimal path element dL
directed radially inward, toward the origin as
shown.
In spherical coordinates, the
correct expression for dL is:
A) d L = + dr r
B) d L = - dr r
C) Neither of these.
cartesian : d L = dx xˆ + dy yˆ
spherical: d L = dr r + r dq qˆ + r sin q df fˆ
dL
x
2.45
Given a spherical SHELL with uniform
surface charge density  (no other charges
anywhere else) what can you say about the
potential V inside this sphere? (Assume as
usual, V(∞)=0)
A) V=0 everywhere inside
B) V = non-zero constant everywhere
inside
C) V must vary with position, but is zero
at the center.
D) Could be A OR B!
E) None of these, it’s something else!
SC-4
We usually choose V(r  ∞) ≡ 0 when
calculating the potential of a point charge to be
V(r) = +κq/r. How does the potential V(r)
change if we choose our reference point to be
V(R)=0 where R is closer to +q than r.
+q
A
B
C
D
R
r
V(r) is positive but smaller than kq/r
V(r) is positive but larger than kq/r
V(r) is negative
V(r) doesn’t change (V is independent of
choice of reference)
∞
+Q
+Q
A
B
Two isolated spherical shells of charge,
labeled A and B, are far apart and each has
charge Q. Sphere B is bigger than sphere A.
Which shell has higher voltage? [V(r=) = 0]
A) Sphere A
B) Sphere B
C) Both have same voltage.
2.46
Why is ò E × dl = 0
in electrostatics?
a) Because Ñ´ E = 0
b) Because E is a conservative field
c) Because the potential (voltage) between
two points is independent of the path
d) All of the above
e) NONE of the above - it's not true!
V
A parallel plate capacitor is attached to a
battery which maintains a constant voltage
difference V between the capacitor plates.
While the battery is attached, the plates are
pulled apart. The electrostatic energy stored in
the capacitor
A) increases
B) decreases
C) stays constant.