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Physics 2113
Jonathan Dowling
Physics 2113
Lecture 07: WED 09 SEP
Electric Fields III
Charles-Augustin
de Coulomb
(1736-1806)
Electric Charges and Fields
First: Given Electric Charges, We
Calculate the Electric Field Using
E=kq/r2.
Charge
Produces EField
Example: the Electric Field Produced
By a Single Charge, or by a Dipole:
Second: Given an Electric
Field, We Calculate the Forces
on Other Charges Using F=qE
Examples: Forces on a Single
Charge When Immersed in the
Field of a Dipole, Torque on a
Dipole When Immersed in an
Uniform Electric Field.
E-Field Then
Produces Force
on Another
Charge
Continuous Charge
Distribution
• Thus Far, We Have Only Dealt With
Discrete, Point Charges.
q
• Imagine Instead That a Charge q Is
Smeared Out Over A:
q
– LINE
q
– AREA
– VOLUME
• How to Compute the Electric Field E?
Calculus!!!
q
Charge Density
• Useful idea: charge density
• Line of charge:
charge
per unit length = 
• Sheet of charge:
 = q/L
 = q/A
charge
per unit area = 
• Volume of charge:
per unit volume = 
charge
 = q/V
Computing Electric Field
of Continuous Charge Distribution
• Approach: Divide the Continuous Charge
Distribution Into Infinitesimally Small
Differential Elements
dq
• Treat Each Element As a POINT Charge
& Compute Its Electric Field
• Sum (Integrate) Over All Elements
• Always Look for Symmetry to Simplify
Calculation!
dq =  dL
dq =  dS
dq = ρ dV
Differential Form of Coulomb’s Law
E-Field
at Point
q2
P1
Differential
dE-Field at
Point
P1
P2
dq2
ICPP: Arc of Charge
• Figure shows a uniformly charged
rod of charge –Q bent into a
circular arc of radius R, centered at
(0,0).
• What is the direction of the electric
field at the origin?
(a) Field is 0.
(b) Along +y
(c) Along -y
y
-
-
-
-
-
-
-
-
-
-
-
x
• Choose symmetric elements
• x components cancel
Arc of Charge: Quantitative
• Figure shows a uniformly charged rod
of charge –Q bent into a circular arc of
radius R, centered at (0,0).
• ICPP: Which way does net E-field
point?
• Compute the direction & magnitude of
E at the origin.
y
–Q
450
x
y dq = l ds = l Rdq
é kdq ù
dEx = [ dE ] cosq = ê 2 ú cosq
ëR û
Ex =
p /2
ò
0
k(l Rdq )cosq kl
=
2
R
R
kl
Ex =
R
dθ
p /2
ò
0
p /2
kl
cosq dq =
sinq
R
0
kl
kl
Enet = 2
Ey =
R
R

Q
Q
2Q
l= =
=
L 2p R / 4 p R
Enet = Ex2 + Ey2
x
(a) toward positive y;
(b) toward positive x;
(c) toward negative y
Charged Ring
The Electric
to a Line
Canceling22-4
Components
- Point Field
P is onDue
the axis:
In the of Charge
Figure (right), consider the charge element on the opposite
side of the ring. It too contributes the field magnitude dE but
the field vector leans at angle θ in the opposite direction from
the vector from our first charge element, as indicated in the
side view of Figure (bottom). Thus the two perpendicular
components cancel. All around the ring, this cancelation
occurs for every charge element and its symmetric partner
on the opposite side of the ring. So we can neglect all the
perpendicular components.
The components
perpendicular to the
z axis cancel; the
parallel components
add.
A ring of uniform positive charge. A differential
element of charge occupies a length ds (greatly
exaggerated for clarity). This element sets up
an electric field dE at point P.
Charged
Ring
22-4 The Electric Field Due to a Line of Charge
Adding Components. From the figure (bottom), we
see that the parallel components each have
magnitude dE cosθ. We can replace cosθ by using
the right triangle in the Figure (right) to write
The components
perpendicular to the
z axis cancel; the
parallel components
add.
A ring of uniform positive charge. A differential
element of charge occupies a length ds (greatly
exaggerated for clarity). This element sets up
an electric field dE at point P.
Charged Ring
22-4 The Electric Field Due to a Line of Charge
Integrating. Because we must sum a huge number
of these components, each small, we set up an
integral that moves along the ring, from element to
element, from a starting point (call it s=0) through the
full circumference (s=2πR). Only the quantity s
varies as we go through the elements. We find
Finally,
The components
perpendicular to the
z axis cancel; the
parallel components
add.
A ring of uniform positive charge. A differential
element of charge occupies a length ds (greatly
exaggerated for clarity). This element sets up
an electric field dE at point P.
ICPP: Field on Axis of Charged Disk
• A uniformly charged circular disk
(with positive charge)
• What is the direction of E at point P
on the axis?
+
+
z
+ + +
+
+ +
+
+
+
(a) Field is 0
(b) Along +z
(c) Somewhere in the x-y plane
P
y
x
Charged Disk is
Integral of Charged
Rings
Q
s=
p R2
dq = s dA = s 2p rdr
Taking R ≫ z gives E-field
above an infinite charged plane:
s
Eplane =
2e 0
A disk of radius R and uniform positive
charge. The ring shown has radius r
and radial width dr. It sets up a
differential electric field dE at point P
on its central axis.
Charged Disk is
Integral of Charged
Rings
Q
s=
p R2
dq = s dA = s 2p rdr
Taking z ≫ R gives E-field
a distance z from a point charge Q:
Epoint
=
ù
ù s é
z
ê1ú
ú=
2
û 2e 0 êë z 1+ ( R / z ) úû
-1/2
ù @ s é1- æ 1- 1 ( R / z )2 ö ù
÷ø ú
úû 2e 0 êë çè 2
û
s é
z
=
ê1- 2
2e 0 ë
z + R2
(
s é
2
1- 1+ ( R / z )
2e 0 êë
)
s R2
Q R2
=
=
2
4e 0 z
4pe 0 R 2 z 2
kQ
= 2 (Couloumb's Law for Point Charge)
z
A disk of radius R and uniform positive
charge. The ring shown has radius r
and radial width dr. It sets up a
differential electric field dE at point P
on its central axis.
Force on a Charge in Electric
Field
Definition of
Electric Field:
Force on
Charge Due to
Electric Field:
Force on a Charge in Electric Field
+++++++++
E
––––––––––
Positive Charge
Force in Same
Direction as EField (Follows)
+++++++++
E
––––––––––
Negative Charge
Force in Opposite
Direction as EField (Opposes)
(a) left
(b) left
-
-
(c) decrease
Electric Dipole in a Uniform Field
• Net force on dipole = 0;
center of mass stays where it
is.
• Net TORQUE  : INTO page.
Dipole rotates to line up in
direction of E.
• |  | = 2(qE)(d/2)(sin )
= (qd)(E)sin
= |p|
E sin
= |p x E|
• The dipole tends to “align”
itself with the field lines.
• ICPP: What happens if the
field is NOT UNIFORM??
Distance Between
Charges = d
-
p = qd
-
+
+
-
-
-
Electric Dipole in a Uniform Field
• Net force on dipole = 0;
center of mass stays where it
is.
• Potential Energy U is smallest
when p is aligned with E and
largest when p anti-aligned
with E.
• The dipole tends to “align”
itself with the field lines.
U = - pE cos0° = - pE
U = - pE cos180° = + pE
Distance Between
Charges = d
-
1 and 3 are “uphill”.
2 and 4 are “downhill”.
U1 = U3 > U2 = U4
U1 = - pE cos (135°) == +0.71pE
U2 = - pE cos ( +45°) == -0.71pE
-
= 45°
U3 = - pE cos ( -135°) == +0.71pE
-
U4 = - pE cos ( -45°) == -0.71pE
t 1 = pE sin ( 45° + 45° + 45°) = pE sin (135°) = 0.71pE
t 2 = pE sin ( 45°) = 0.71pE
t 3 = pE sin ( -135°) = 0.71pE
t 4 = pE sin ( -45°) = 0.71pE
t1 = t 2 = t 3 = t 4
(a) all tie;
(b) 1 and 3 tie, then 2 and 4 tie
Summary
• The electric field produced by a system of charges
at any point in space is the force per unit charge
they produce at that point.
• We can draw field lines to visualize the electric
field produced by electric charges.
• Electric field of a point charge: E=kq/r2
• Electric field of a dipole:
E~kp/r3
• An electric dipole in an electric field rotates to
align itself with the field.
• Use CALCULUS to find E-field from a continuous
charge distribution.