Transcript Physics 2102 Spring 2002 Lecture 2

Physics 2102
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Jonathan Dowling
Physics 2102
Lecture: 03 TUE 26 JAN
Electric Fields II
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Michael Faraday
(1791-1867)
What Are We Going to Learn?
A Road Map
• Electric charge
- Electric force on other electric charges
- Electric field, and electric potential
• Moving electric charges : current
• Electronic circuit components: batteries, resistors,
capacitors
• Electric currents - Magnetic field
- Magnetic force on moving charges
• Time-varying magnetic field Electric Field
• More circuit components: inductors.
• Electromagnetic waves - light waves
• Geometrical Optics (light rays).
• Physical optics (light waves)
Coulomb’s Law
 q1
F12
q2
F21
r12
k | q1 | | q2 |
| F12 |
2
r12
For Charges in a
Vacuum
k
2
N
m
9
8
.
99

10
=
C2
Often, we write k as:
k 1
4 0
with  0  8.85 10
12
2
C
N m2
E-Field is E-Force Divided by ECharge
r
r F
Definition of
E

Electric Field:
q
r
k | q1 | | q2 |
| F12 |
2
r12

r
k | q2 |
| E12 |
2
r12
EForce
on
Charge
+q1
P1
P1

–q2
P2
r
E12
E-Field

at Point
Units: F = [N] = [Newton] ;
r
F12
–q2
P2
E = [N/C] = [Newton/Coulomb]
Force on a Charge in Electric
Field
Definition of
Electric Field:
Force on
Charge Due
to
Electric Field:
r
r F
E
q
r
r
F  qE
Force on a Charge in Electric
Field
+++++++++
Positive Charge
Force in Same
Direction as EE
Field
(Follows)
––––––––––
+++++++++
E
––––––––––
Negative Charge
Force in Opposite
Direction as EField (Opposes)
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.
• What happens if the field is
NOT UNIFORM??
Distance Between
Charges = d
Electric Charges and Fields
First: Given Electric Charges, We
Calculate the Electric Field Using
E=kqr/r3.
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
charge
 = q/A
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 =  dV
Differential Form of Coulomb’s
Law
r
k q2
E12  2
r12
E-Field
at Point
r
E12
P1
q2
P2

r
k dq2
dE12 
2
r12
Differential
dE-Field at
Point

r
dE12
P1
dq2
Field on Bisector of Charged Rod
• Uniform line of charge
+q spread over length L
• What is the direction of
the electric field at a
point P on the
perpendicular bisector?
(a) Field is 0.

(b) Along +y
(c) Along +x
• Choose symmetrically
located elements of length
dq = dx
• x components of E cancel
E

r
dE
s
dE
P
y
x
dx
a
q
o
L
dx
Line of Charge: Quantitative
• Uniform line of charge,
length L, total charge q
• Compute explicitly the
magnitude of E at point
P on perpendicular
bisector
• Showed earlier that the
net field at P is in the y
direction — let’s now
compute this!
P
y
a
x
q
o
L
Line Of Charge: Field on bisector
Distance hypotenuse:
dE
Charge per unit length:
P
q

L

2 1/2
[C/m]
k(dq)
dE 
r2
a
r

r a x
2
dx
q
k ( dx)a
dE y  dE cos  2
(a  x 2 )3 / 2
x o
L
a
a
cos   2
r (a  x 2 )1/2
Adjacent
Over
Hypotenuse
Line Of Charge: Field on bisector
L/2
L/2
dx


x
E y  k a 
2
2 3 / 2  k a  2
2
2
(
a

x
)
 a x  a  L / 2
L / 2
Integrate: Trig Substitution!
Point Charge Limit: L << a
2kL
kL kq
Ey 
 2  2
2
2
a
a
a 4a  L

2kL
a 4a 2  L2
Line Charge Limit: L >> a
2kL
2k
Ey 

a
a 4a 2  L2
Units Check!
Coulomb’s
Law!
Nm2 1 C  N 

  
 C m m m
Binomial Approximation from Taylor Series:
x<<1
1 x
1 nx
n
2 1/ 2
2kL
kL   L  
 2 1   
2
2
a  2a  
a 4a  L
 2kL
2 



kL
1 L
kL
 2 1    2 ; (L  a)
a  2 2a   a
1/ 2
2
2kL  2a  

1   

aL   L  
a 4a 2  L2
2
2k  1 2a   2k

; (L  a)
1   
a  2  L   a
Example — Arc of Charge:
Quantitative y
• Figure shows a uniformly
charged rod of charge -Q bent
into a circular arc of radius R,
centered at (0,0).
• Compute the direction &
magnitude of E at the origin.
kdQ
dE x  dE cos  2 cos
R
 /2
 /2
k (Rd ) cos k
Ex 

0
k
Ex 
R
R
2

k
Ey 
R
R
–Q
E
450
x
y
 cosd
dQ = Rd
d

0
k
E 2
R
x
2Q/(R)
E  Ex2  Ey2
Example : 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?
(a) Field is 0
(b) Along +z
(c) Somewhere in the x-y plane
P
z
y
x
Example : 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
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.