31
Electric Lecture Fields
ACT: Two balls
Two balls of equal mass are suspended from the ceiling with nonconducting wire. One ball is given a charge q and the other is given a charge 3q. Which of the following best represents the equilibrium position?
3q
q q
3q
3q
q
A
B
C
Same force on both (3rd law) and same mass!
What is a "field"?
Very basic definition:
Assignment of a scalar or a vector to each point in space.
Example of scalar field: Temperatures
7 7 73 72 71
82 84 83
75
68 64 80 73 82 88 55 66 88 75 80 88 83 90 91 92
77
The shown temperatures are samples of the field
Example of vector field: Winds
77 73 72 71
82 84 83
88
75
77 68 64 80 73 57 56 55 66 80 88 75 90 83 92 91
The shown temperatures are samples of the field
Electric field
Definition:
If the electric force on a test charge q located at point P is F, then the electric field at point P is:
E
F q
Units: N/C
Because the force is always proportional to q, the electric field is independent of the test charge!
Electric field by a point charge
The electric force by Q on q is
F
Q
+
q
+
E
F
Qq ^ k 2 r r
Independent of q (test charge)
Q
+
F q
E
The electric field by Q (source charge) is:
E
F q
E
Q ^ k 2r r
Electric field by a negative/positive point charge
+
-
If you place a test charge q near the source charge, you get a force.
+
q
If you place a test charge q in a different place, you might get a different force.
+
q
If the test charge is 2q, the force is twice as large:
+
2q
Positive charges feel a force in the same direction as the electric field. Negative charges feel a force in the opposite direction as the electric field.
-q < 0
+
q>0
But none of this affected or changed the field, which only depends on the source charge!
+
Relate to: ACT: "g due to two stars", lecture 29. g only depends on source mass
Superposition
For several point-like charges or for continuous charge distributions, we use the superposition principle: Just as the net force is the vector-sum of all the forces, the net electric field is the vector-sum of the electric fields by all the parts.
ACT: Net electric field
Which of the three vectors best represents the direction of the net electric field at the location of charge Q?
q1 < 0
Enet
A
E1
q2 > 0
Q<0
B
E2
C
Continuous charge distributions
Each small differential charge dq is like a point charge. dq (r )dV
P The net E-field at point P is the vector sum (integral) of the electric field contributions from all of them:
dE
dq ^ ke 2 r r
E
dq ^ ke 2 r r enti re volume
EXAMPLE: Ring
Below is a ring of radius R with uniform linear charge distribution . Find the electric field along the z-axis.
x
P
z
y
x
dq = Rd
d
r
R2
z2
z
P
y
dE
dq k 2 r
k
R2
Rd
z2
x
Cylindrical symmetry: only Ez matters
P
z
dEz
k
dE cos
R2 Rd z2 R2 zRd z z2
y
k
R
zR z
2
z
2 3/ 2
E z , net
k
zR R
2
2 0
z
2 3/ 2
d
k
2
R
2
2 3/ 2
From the ring to the charged disk...
Disk of radius R with uniform charge surface density can be thought of as made of rings of radii r = 0 to r = R and linear charge density dr.
E disk R
R
0
k
2
zr z
1
2 3/ 2
r
2
dr
1
k2
z
1
R
r
1
2
z
2 1/2 0
k2
z
r
2
z
2 1/2
z
k2
z R
2
z
2 1/2
2
1
0
z R
2
z
2 1/2
... and to the infinite plane
Take the limit R
R
lim
2
1
0
z R
2
z
2 1/2
2
0
Uniform electric field ! (Does not depend on distance)
Example: Electron in a uniform electric field
Once the electric field is known, finding the force on a given charge is trivial...
Example: Describe the motion of an electron with initial velocity v ^ that enters a region with a uniform electric field E E 0 j .
v 0i^
a
Fnet me
Fe me
qeE me
.
qe E 0 ^ j me
Constant acceleration in the j direction. Identical to projectile motion!
v0
electron
E
F = |qe|E
parabola
Electric field in conductors
The electric field inside a conductor in equilibrium is always zero.
If E 0
F
0
a
0
motion of charges (conductor, charges can move) non equilibrium
The electric field right outside a conductor in equilibrium are perpendicular to the surface of the conductor.
We cannot have a force parallel to the surface (would produce motion), but perpendicular to it is OK.
E=0
Electric field in insulators (or dielectrics)
Charges can move only a little (polarization), so the electric field is not zero, but it is smaller than outside the insulator.
Enet
E ext
+
Epol
Eext
+ +
Enet Eext
+ +
+
Eext
+
+
+
+
Epol
Dielectric breakdown
As Eext increases, the electric force that pulls apart the + and parts of each molecule in the insulator increases. Insulator Insulator
+ +
+ ++
Eext 0
Eext
Eext = 0
stronger
0,
Insulator
+ ++
Dielectric breakdown!
When the electric force is stronger than the molecular bond, the molecule breaks apart.
Conductor
+ ++
Eext > EMAX
Dielectric strength
EXAMPLE: Lightning
Collisions inside clouds produce charged particles. The heavier particles are () and sit at the bottom of the cloud and the lighter particles (+) near the top.
- - - - -
The negative bottom attracts the positives on the ground.
- -
-
- -
- - - -- - -
Factoids:
When E > EMAX for air, air becomes a conductor
EMAX air ~ 3x106 N/C
t ~ 30 ms
V ~ 20x106 V
E
I ~ 40,000 A
P ~ 1012 W
+ + + + + + + + +
DEMO: Lightning and fluorescent
EXAMPLE: Infinite line of charge
A cable of diameter D = 3 mm and length L = 200 m has a total charge Q = 4 C uniformly distributed along its length. Find the electric field at point P, located at a distance r = 2 cm from it.
P
r
L >> r infinite cable
r >> D
one-dimensional charge distribution
y
dE
P
dq ke 2 d
dx ke 2 x r2
dE r
d
x2
r2 dq = dx
x
x
dx
Linear charge density:
Q L
Infinite line: There is always another dq symmetrically placed. The x-components of the electric fields cancel out.
dE y
dx ke 2 cos 2 x r y
dx r ke 2 x r2 x2 r2
ke
r dx x
2
r
2 32
dE r
dE
d x2 r2
x
x
Enet
dE y
ke r
dx x
2
r
2 32
...
2ke
r
Q 2ke rL