Class 02: Outline
Answer questions
Hour 1:
Review: Electric Fields
Charge
Dipoles
Hour 2:
Continuous Charge Distributions
P02 - 1
Last Time: Fields
Gravitational & Electric
P02 - 2
Gravitational & Electric Fields
CREATE:
FEEL:
Mass M
Charge q (±)
G
M
g = −G 2 rˆ
r
G
q
E = ke 2 rˆ
r
G
G
Fg = mg
G
G
FE = qE
This is easiest way to picture field
P02 - 3
PRS Questions:
Electric Field
P02 - 4
Electric Field Lines
1. Direction of field line at any point is tangent to
field at that point
2. Field lines point away from positive charges
and terminate on negative charges
3. Field lines never cross each other
P02 - 5
In-Class Problem
P
ĵ
s
−q
d
î
+q
Consider two point charges of equal magnitude but
opposite signs, separated by a distance d. Point P
lies along the perpendicular bisector of the line
joining the charges, a distance s above that line.
What is the E field at P?
P02 - 6
Two PRS Questions:
E Field of Finite
Number of Point
Charges
P02 - 7
Charging
P02 - 8
How Do You Charge Objects?
• Friction
• Transfer (touching)
• Induction
+q
-
Neutral
+
+
+
+
P02 - 9
Demonstrations:
Instruments for
Charging
P02 -10
Electric Dipoles
A Special Charge Distribution
P02 -11
Electric Dipole
Two equal but opposite charges +q and –q,
separated by a distance 2a
G
p
Dipole Moment
q
2a
-q
G
p
G
p ≡ charge×displacement
= q×2aˆj = 2qaˆj
points from negative to positive charge
P02 -12
Why Dipoles?
Nature Likes To Make Dipoles!
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/20-Molecules2d/20-mole2d320.html
P02 -13
Dipoles make Fields
P02 -14
Electric Field Created by Dipole
Thou shalt use
components!
G
rˆ
r
∆x ˆ ∆y ˆ
j
=
=
i
+
2
3
3
3
r
r
r
r
⎛ ∆x ∆x ⎞
Ex = keq⎜ 3 − 3 ⎟
⎝ r+ r− ⎠
⎛
⎞
x
x
⎟
= keq⎜
−
⎜ ⎡x2 + ( y − a)2 ⎤3/2 ⎡x2 + ( y + a)2 ⎤3/2 ⎟
⎦
⎣
⎦ ⎠
⎝⎣
⎛
⎞
⎛ ∆y+ ∆y− ⎞
y −a
y+a
⎜
⎟
−
Ey = keq ⎜ 3 − 3 ⎟ = keq
2
2 3/ 2
2
2 3/ 2 ⎟
⎜
r
r
⎡
⎤
⎡⎣ x + ( y + a) ⎤⎦
− ⎠
⎝ +
⎝ ⎣ x + ( y − a) ⎦
⎠
P02 -15
PRS Question:
Dipole Fall-Off
P02 -16
Point Dipole Approximation
Take the limit r >> a
Finite Dipole
You can show…
3p
Ex →
sin θ cos θ
3
4πε 0 r
Ey →
Point Dipole
p
4πε 0 r
( 3cos θ − 1)
2
3
P02 -17
Shockwave for Dipole
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/06DipoleField3d/06-dipField320.html
P02 -18
Dipoles feel Fields
P02 -19
Demonstration:
Dipole in Field
P02 -20
Dipole in Uniform Field
G
E = Eˆi
G
p = 2qa (cos θ ˆi + sin θ ˆj)
G
G G
G
G
Total Net Force: Fnet = F+ + F− = qE + (−q)E = 0
Torque on Dipole:
G G G G G
τ = r×F = p×E
τ = rF+ sin(θ ) = ( 2a )( qE ) sin(θ ) = pE sin(θ )
G
p
tends to align with the electric field
P02 -21
Torque on Dipole
Total Field (dipole + background)
shows torque:
http://ocw.mit.edu/ans7870/8/
8.02T/f04/visualizations/electr
ostatics/43torqueondipolee/43torqueondipolee320.html
• Field lines transmit tension
• Connection between dipole field and
constant field “pulls” dipole into alignment
P02 -22
PRS Question:
Dipole in Non-Uniform Field
P02 -23
Continuous Charge
Distributions
P02 -24
Continuous Charge Distributions
Break distribution into parts:
V
Q = ∑ ∆ qi → ∫ dq
i
V
E field at P due to ∆q
G
G
∆q
dq
∆ E = ke 2 rˆ → d E = ke 2 rˆ
r
r
Superposition:
G
E( P) = ?
G
G
G
E = ∑ ∆E → dE
∫
P02 -25
Continuous Sources: Charge Density
R
Volume = V = π R 2 L
L
w
dQ = ρ dV
Q
ρ=
V
L
dQ = σ dA
Q
σ=
A
Length = L
dQ = λ dL
Area = A = wL
L
Q
λ=
L
P02 -26
Examples of Continuous Sources:
Line of charge
dQ = λ dL
Length = L
Q
L
λ=
L
http://ocw.mit.edu/a
ns7870/8/8.02T/f04
/visualizations/elect
rostatics/07LineIntegration/07LineInt320.html
P02 -27
Examples of Continuous Sources:
Line of charge
dQ = λ dL
Length = L
Q
L
λ=
L
http://ocw.mit.edu/a
ns7870/8/8.02T/f04
/visualizations/elect
rostatics/08LineField/08LineField320.html
P02 -28
Examples of Continuous Sources:
Ring of Charge
Q
λ=
dQ = λ dL
2π R
http://ocw.mit.edu/a
ns7870/8/8.02T/f04
/visualizations/elect
rostatics/09RingIntegration/09ringInt320.html
P02 -29
Examples of Continuous Sources:
Ring of Charge
Q
λ=
dQ = λ dL
2π R
http://ocw.mit.edu/a
ns7870/8/8.02T/f04
/visualizations/elect
rostatics/10RingField/10ringField320.html
P02 -30
Example: Ring of Charge
P on axis of ring of charge, x from center
Radius a, charge density λ.
Find E at P
P02 -31
Ring of Charge
1) Think about it
E⊥ = 0 Symmetry!
http://ocw.mit.edu/a
ns7870/8/8.02T/f04
/visualizations/elect
rostatics/09RingIntegration/09ringInt320.html
2) Define Variables
dq = λ dl = λ ( a dϕ )
r= a +x
2
2
P02 -32
Ring of Charge
3) Write Equation
G
G
rˆ
r
dE = ke dq 2 = ke dq 3
r
r
dq = λ a dϕ
r = a2 + x2
a) My way
x
dEx = ke dq 3
r
b) Another way
G
1 x
x
dEx = dE cos(θ ) = ke dq 2 ⋅ = ke dq 3
r r
r
P02 -33
Ring of Charge
dq = λ a dϕ
4) Integrate
r = a2 + x2
x
Ex = ∫ dEx = ∫ ke dq 3
r
x
= ke 3 ∫ dq
r
Very special case: everything except dq is constant
∫ dq = ∫
2π
0
2π
λ a dϕ = λ a ∫ dϕ = λ a 2π
0
=Q
P02 -34
Ring of Charge
5) Clean Up
x
E x = ke Q 3
r
E x = ke Q
G
E = ke Q
x
(a
2
+x
2
)
3/ 2
6) Check Limit a → 0
x
(a
2
+x
2
)
3/ 2
ˆi
E x → ke Q
x
(x )
2
3/ 2
ke Q
= 2
x
P02 -35
In-Class: Line of Charge
ĵ
r̂
P
î
s
−
L
2
+
L
2
Point P lies on perpendicular bisector of uniformly
charged line of length L, a distance s away. The
charge on the line is Q. What is E at P?
P02 -36
Hint: http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/07-LineIntegration/07LineInt320.html
ĵ
r̂
θ
î
P
r = s 2 + x ′2
s
θ
−
L
2
x′
dq = λ dx ′
dx ′
+
L
2
Typically give the integration variable (x’) a “primed”
variable name.
P02 -37
E Field from Line of Charge
G
Q
ˆj
E = ke
2
2
1/ 2
s ( s + L / 4)
Limits:
G
Qˆ
lim E → ke 2 j
s >> L
s
G
Q ˆ
λˆ
j = 2ke j
lim E → 2ke
s << L
Ls
s
Point charge
Infinite charged line
P02 -38
In-Class: Uniformly Charged Disk
(x > 0)
P on axis of disk of charge, x from center
Radius R, charge density σ.
Find E at P
P02 -39
Disk: Two Important Limits
⎡
G
σ ⎢
x
1−
Edisk =
2
2
2ε o ⎢
x
R
+
⎣
(
Limits:
G
***
E
→
lim
disk
x >> R
1
Qˆ
i
2
4πε o x
G
σ ˆ
i
lim Edisk →
x << R
2ε o
)
1/ 2
⎤
⎥ ˆi
⎥
⎦
Point charge
Infinite charged plane
P02 -40
E for Plane is Constant????
1)
2)
3)
4)
Dipole:
E falls off like 1/r3
Point charge: E falls off like 1/r2
Line of charge: E falls off like 1/r
Plane of charge: E constant
P02 -41