chapter 21 electric potential
DESCRIPTION
Conservation of energy Conservative force Conservation of energy + charged Potential (energy) air → conductor Potential difference (voltage) discharge 2TRANSCRIPT
Chapter 21 Electric Potential
++ + +
2
Potential difference(voltage)
air → conductor
discharge
charged
Conservative force
Conservation of energy
Potential (energy)
Coulomb force is conservative
3
1 22
m mF Gr
Comparing : 1 22~ Q QF k
r
Electrostatic / Coulomb force is conservative
Work doesn’t depend on the path:
1 2L L
W qE dl qE dl
0E dl
The circulation theorem of electric field
( 0 )E
Electric potential
4
Conservative force → potential energy U
+
-
U depends on the charge!
Define electric potential:
/V U q
Potential difference / voltage:
a b abab a b
U U WV V Vq q
Electric potential & electric field
5
Difference in potential energy between a and b:b
b a aU U qE dl
If position b is the zero point:
ab
b
ba b a aV V V E dl
Relationship between electric field and potential
0V
a aV E dl
Unit of electric potential: Volt (V)
Zero point
6
Electric potential:
It depends on the chosen of zero point
0V
a aV E dl
Usually let V = 0 at infinity: a aV E dl
For the field created by a point charge :
Q.a
204a a
QV drr
04 a
Qr
Properties of electric potential
7
1) Va → potential energy per unit (+) charge
b
ba b a aV V V E dl
3) Va and Vba are scalars, differ from E vector
0V
a aV E dl
2) Va depends on the zero point, Vba does not
0,
V
a a aU qV q E dl
b
ab ab aW qV q E dl
Calculation of electric potential
8
Potential of point charge:
04QV
r (V = 0 at
infinity)System of several point charges:
04i
i
QVr
0
or 4
dQVr
If the electric field is known:0V
a aV E dl
dQ
r.A
Potential of electric dipole
9
Example1: (a) Determine the potential at o, a, b. (b) To move charge q from a to b, how much work must be done?
Q Qo ..
a.h
l lb
Solution: (a)0 0
04 4o
Q QVr r
, 0aV
0 04 8bQ QV
l l
08Q
l
(b)08ba b a
QV V Vl
0
8baQqW qV
l
Charged conductor sphere
10
Example2: Determine the potential at a distance r from the center of a uniformly charged conductor sphere (Q, R) for (a) r > R; (b) r = R; (c) r < R.
Solution: All charges on surface: ++
+
++
++
++
+ Q2
0
( ),4
QE r Rr
0 ( )E r R
(a) V at r > R:
204a r
QV drr
04Q
r Same as point charge
.a
11
++
+
++
++
++
+ Q
(b) V at r = R:
20 04 4b R
Q QV drr R
(c) V at r < R: b
c bcV E dl V
04bQV
R
.b
.c
Discussion:
(1) Conductor: equipotential body
(2) Charges on holey conductor
+ + +
--- ?
Same potential at any point!
Uniformly charged rod
12
Example3: A thin rod is uniformly charged (λ, L). Determine the potential for points along the line outside of the rod.
Solution: Choose infinitesimal charge dQ
xa
dL
dx
dQ0
ln4
d Ld
aV d L
d
04
dxx
.b
Potential at point b?
Charged ring
13
Example4: A thin ring is uniformly charged (Q, R). Determine the potential on the axis.
Solution:
R
xQ
. a
r
dQ
aV Ring
04dQ
r 04Q
r
or:2 2
0
4
aQVx R
Discussion:
(1) Not uniform? (2) x >> R (3) other shapes
Conical surface
14
Question: Charges are uniformly distributed on a conical surface (σ, h, R). Determine the electric potential at top point a. (V = 0 at infinity)
h
R
a
Electrostatic induction
15
Question: Put a charge q nearby a conductor sphere initially carrying no charge. Determine the electric potential on the sphere.
R
b
qo
----
-
-
++
+
+
+
+
Q
Connect to the ground?
*Breakdown voltage
16
Air can become ionized due to high electric field
Air → conducting → charge flows
Breakdown of air: 63 10 /E V m
Breakdown voltage for a spherical conductor?
Q
R04 R
QV E RR
5cm 15000VR V
Equipotential surfaces
17
Visualize electric potential: equipotential surfaces
Points on surface → same potential
(1) Surfaces ⊥ field at any point.
(3) Move along any field line, V ↘
(4) Surfaces dense → strong field
(2) Move on surface, no work done
Some examples
18
(1) Equipotential surfaces for dipole system:
(3) Parallel but not uniform field lines?
(2) Conductors
E determined from V
19
To determine electric field from the potential, useb
b a aV V E dl
In differential form: dV E dl
lE dl
Component of E in the direction of dl :
/lE dV dl
Total electric field: V V VE i j kx y z
Gradient of V
20
Gradient of V :
Gradient → partial derivative in direction V changes most rapidly
dVV ndn
n
V
V+dV
1 2
dl
dna b
E
ldVEdl
dVEdn
V V VE V i j kx y z
a
V E dl
Determine E from V
21
Example5: The electric potential in a region of space varies as V = x2yz. Determine E.
Solution:x
VEx
2 ,xyz
yVEy
2 ,x z
2 2 2E xyzi x zj x yk
zVEz
2x y
Determine V from E
22
Solution: 2xVE xyx
2V xydx 2 ( )x y C y
Example6: The electric field in space varies as . Determine V.2 22 ( )E xyi x y j
2 ( )y
V dC yE xy dy
2 2x y
313
y C
2 31 3
V x y y C ( C is constant )