Download lectures from Chapter 27

How to Calculate the Electric
Field of Objects
(Chapter 27)
The Electric Field of a Collection of Point
Charges:
The net electric field due to a group of point charges is
where Ei is the field from point charge i.
Remember:
This is a vector sum!
You do not add
Magnitudes.
E-Field of Two Point Charges
The electric field of a point charge q at the origin, r = 0, is
where є0 = 8.85 × 10–12 C2/N m2 is the permittivity
constant.

q1
r1

q2
r2
!"
E net =
1 # q1 # q2 # &
r + 2 r2 (
2 1
%
4!" 0 $ r1
r2 '
General Principles of Electric Field Lines
• Alternate way to visualize the E-field vector at each
point.
• E-field vectors are tangent to the field lines at all
points.
• E-field lines point away from positive charges and
towards negative charges.
• Density/spacing of field lines=E field magnitude
• E-field lines NEVER cross each other!!! (Why ???)
Electric field lines: Two Point Charges
Compare and contrast dipole and two like charges.
Limiting Cases (Important)
Limiting cases:
A limiting case is the behavior of a physical system when one of
the parameters is set to an extreme value.
In such cases, complex systems have very simple forms that are easy
to analyze mathematically.
Example: Localized distribution of charges
--->How does it behave at large distances ????
Example: The electric field of 3 point charges
(textbook)
(Enet )x = (E1 )x + (E2 )x + (E3 )x = 2(E1 )x + (E2 )x
E-fields of the charges…
1 q2
(E2 ) x = E2 =
4!" 0 x 2
1
xq
(E1 )x = E1 cos# =
4!" 0 (x 2 + d 2 )3/2
(Enet )x =
q $ 1
2x
'
+
4!" 0 &% x 2 (x 2 + d 2 )3/2 )(
As x goes to infinity, the electric field vanishes. (Wrong limit!)
Instead let x>>d,
1 3q
Enet (x >> d) =
4!" 0 x 2
The limiting case is
a point particle.
Electric Dipoles
•
There are two basic types of dipoles
– Permanent dipoles such as polar molecules (water)
– Induced dipoles, atoms or molecules that are polarized by
an electric field
• Induced dipoles are important in optics. Optical properties
(reflection, refraction, absorption, emission, scattering) of
materials due to induced dipoles
• Radio antennas are often designed as electric dipoles that
oscillate when connected to an AC current producing
“dipole radiation”.
Polar covalent molecules (eg. water)
dipole antenna
It’s All About the Dipole Moment
We can represent an electric dipole by two opposite charges
±q separated by the small distance 2a.
The Dipole-Moment
!
p = charge ! displacement
!
p = points from negative to positive charge
!
p=2qaĵ
The electric field can be analyzed as the vector
sum of two point charges
Calculating the E-field of a Finite Dipole
!
" +q
!q %
E = ke $ 2 r̂+ + 2 r̂! '
r! &
# r+
Limit of an “Ideal” Dipole
• “Ideal dipole” obtained
by taking the limit r>>a
• E-field of ideal dipole
depends only on p
Finite “Real” Dipole
Limit of an ideal dipole:
Point “Ideal” Dipole
Full Expression for Electric Field
!"
E dipole
!"
!"
1 3( pir# )r# # p
=
3
4!" 0
r
!"
( pir# )r# is the component of the dipole moment along the direction of
the radial unit vector from the origin
The electric field decreases like 1/r3 instead of 1/r2 for a point
particle.
Why does it fall off faster than that of a point particle ?
What kind of force does a dipole feel in an
electric field ????
Dipoles in an Electric Field
In a constant electric field there is no net force on the dipole.
But there is a torque !
Torque is
‘rotational force’:
! ! "!
! =r"F
Torque of a Dipole in Constant E Field:
Electric Fields Produced by Continuous Charge
Distributions: Mathematical Summary
1
ke =
4!" 0
Uniformly Charged Wires and Rods
The linear charge density of an
object of length L and charge
Q, is defined as
Electric Field of Finite
Length Uniformly Charge
Rod/Wire
Rod is in x-y plane. By
symmetry Ez=0
Solving using Calculus
Step 1: General Formula for Charge Distribution
Step 2: Express r and dq in terms of y and !. Change integration
variable to !.
r = y / cos! ,
dq = " dx = " (dx / d! )d! = d!" y / cos 2 ! using x = y tan !
r! = cos! !j # sin ! !i
An Infinite Line of Charge
We can obtain the field for an infinite line of charge is x1 and x2
go to inifinity
tan !1 = x1 / y
tan! 2 = x2 / y
lim !1 = $% / 2 lim ! 2 = % / 2
x1 "#
x2 "#
Inserting into the electric fields…
2k ! 2kQ
Ey = k(! / y)(+1 " ("1)) =
=
y
Lr
Ex = k(! / y)(0 " 0) = 0
Knight obtains the same answer by different means…
Where r is the radial distance away from the rod.
Limiting Cases: Electric Field of Point Charge,
Finite Line Segment, and Infinite Line Charge
Very close to the line segment, it’s E-Field
is indistinguishable from infinite line charge
Very far away,
the line segment
looks like a point
charge
A Ring of Charge
P is on the axis of the ring
at a distance x from center.
Want to find E-field at P
Setting up the problem…
Because of the symmetry
with respect to rotations
around the ring axis, the
field perpendicular to axis
vanishes,
E! = 0
dq = ! dl = ! (ad" )
r= a +x
2
2
Adding up the E-field…
Step 1: E-field due to single charge element dq
Step 2: Integrate dE with respect to dq around the ring.
Because of the symmetry
of P on the ring axis,
x and r do not depend
on the position of the
charge element dq.
Finalize the solution (ring of charge)
Step 3: Note that the integral of dq around the ring is the total
charge of the ring, Q
Step 4: Put all of the pieces together
What is the limiting
case ??
A Disk of Charge
We can obtain the E-field
for a disk by dividing
the disk into infinitesimal
rings of charge and use
our result for a ring of
charge.
Q
Q
!= =
2
A "R
Calculating the E-field for a Disk
Step 1: A ring of radius r has a thickness dr and charge
dQ = 2!"rdr
Step 2: Electric field due to the ring on axis is
dEz = k
(z
zdQ
2
+r
)
2 3/2
Step 3: Integrate the E-fields for all ring segments comprising
the disk
R
R
Ez = ! dEz = !
0
0
(z
kz
2
+r
)
2 3/2
2"#r dr
%
(
z
Ez = 2" k# ' 1 $
2
2 *
&
z +R )
E-field for in Infinite Sheet of Charge
If we let the size of the disk become infinite, we obtain
the E-field for an infinite plane of charge
( Edisk )z
&
)
z
"
= 2! k" lim ( 1 %
= 2! k" =
2
2 +
R#$
2, 0
'
z +R *
E field does not decrease with distance ! How can this
be a correct limiting case ???
A Plane of Charge
">0, E points away from charge
sheet.
"<0, E points toward the charge
sheet.
E-field dependence on distance (x)
for finite disk, infinite sheet, and point charge
Disk size=R=5cm
Summary of different charge distribution:
Dependence on distance r
•
•
•
•
Electric Dipole: E-field falls off like 1/r3
Point Charge: E-field falls off like 1/r2
Infinite line of charge: E-field falls off like 1/r
Infinite plane/sheet of charge: E-field is constant !
The Parallel-Plate Capacitor
Two electrodes, one with
charge +Q and the other
with –Q placed face-toface a distance d apart.
Capacitors: Critical to Technology
• Capacitors are one type of passive components used in electrical
circuits. Other passive components are resistors, inductors, and
diodes. Unlike active components (transistors, op-amps, logic
gates), passive components have a single input and output lead.
• Capacitors store electrical energy that can later be released very
quickly
• Capacitors are essential elements along with resistors and
inductors for constructing analog filters and electrical oscillators.
Uses for Capacitors
•
•
•
•
•
•
•
•
•
•
Temporary energy storage (eg. to prevent loss of data when changing batteries
of a device)
Pulsed power: Store and release energy in pulses to power devices like high
power pulsed lasers ,camera flashes.
Computer memory: DRAM memory stores each bit in the form of charge on a
capacitor
Digital Cameras: CCD image sensors use a capacitor array to store the
captured image as electrical charges.
Power conditioning: Capacitors are used to smooth out current fluctuations
from AC wall current to provide smooth steady currents for other circuits
(such as audio equipment, computers)
Signal processing: Capacitors are essential for analog filters used to remove
high frequency noise from a AC signal, block DC components of AC signals,
or as radio tuners
Microphones (Condenser microphones are a variable capacitor)
Fingerprint sensors
Airbag sensors rely on a mechanical variable capacitor (MEMS).
The iPhone touch screen is a pressure sensitive capacitor.
Condenser Microphone
iPhone
Touch
Screen
But it doesn’t look like a parallel plate
capacitor…
An early parallel plate
capacitor from the 1800’s
Capacitors in modern circuits
don’t look like the parallel plates
in the textbook. Why ?
Types of Capacitors
Parallel-Plate Capacitor
Cylindrical Capacitor
A cylindrical capacitor is a parallel-plate capacitor that has
been rolled up with an insulating layer between the plates.
L
d
The ideal capacitor assumes that the plates are infinite in size.
Real capacitors have finite plates and non-uniform‘fringe fields’.
When can we approximate a real capacitor to be ideal ???
Deriving the electric field
Ecapacitor
!
Q
=
=
"0 "0 A
Motion of a Charged Particle in an Electric Field
The electric field exerts a force
on a charged particle. If this is the only force acting on q, it
causes the charged particle to accelerate with
Two particles with the same q/m
have the exact same trajectories!
In a uniform field, the acceleration is constant:
!
d x q !
= E
2
dt
m
2
1 " q !% 2 !
!
!
! x(t) = $ E ' t + v(0)t + x(0)
2# m &