Physics 2220
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Transcript of Physics 2220
Physics for Scientists and Engineers II , Summer Semester 2009
Physics 2220
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
Chapter 23: Electric Fields
• Materials can be electrically charged.• Two types of charges exist: “Positive” and “Negative”.• Objects that are “charged” either have a net “positive” or a net “negative”
charge residing on them.• Two objects with like charges (both positively or both negatively charged)
repel each other.• Two objects with unlike charges (one positively and the other negatively
charged) attract each other.• Electrical charge is quantized (occurs in integer multiples of a fundamental
charge “e”).q = N e (where N is an integer)electrons have a charge q = - e protons have a charge q = + eneutrons have no charge
Physics for Scientists and Engineers II , Summer Semester 2009
Material Classification According to Electrical Conductivity
• Electrical conductors: Some electrons (the “free” electrons) can move easily through the material.
• Electrical insulators: All electrons are bound to atoms and cannot move freely through the material.
• Semiconductors: Electrical conductivity can be changed over several orders of magnitude by “doping” the material with small quantities of certain atoms, making them more or less like conductors/insulators.
Physics for Scientists and Engineers II , Summer Semester 2009
Shifting Charges in a Conductor by “Induction”
+
++
+
+
-
-
-
-
-
---
Negatively charged rod
uncharged metal sphere
+
++
+
+ --
-
-
-
---
Left side of metal spheremore positively charged
Right side of metal spheremore negatively charged
Physics for Scientists and Engineers II , Summer Semester 2009
Coulomb’s Law (Charles Coulomb 1736-1806)
Magnitude of force between two “point charges” q1 and q2 .
2
21
r
qqkF ee
2
212
2
29
102854.8
4
1106987.8
mN
Cxwhere
C
Nmxk
o
oe
r = distance betweenpoint charges
Coulomb constant
Permittivity of free space
Physics for Scientists and Engineers II , Summer Semester 2009
Charge
Unit of charge = Coulomb
Smallest unit of free charge: e = 1.602 18 x 10-19 C
Charge of an electron: qelectron = - e = - 1.602 18 x 10-19 C
Physics for Scientists and Engineers II , Summer Semester 2009
Vector Form of Coulomb’s Law
Force is a vector quantity (has magnitude and direction).
12221
12 r̂r
qqkF e
unit vector pointing fromcharge q1 to charge q2
Force exerted by charge q1 on charge q2
(force experienced by charge q2 ).
Physics for Scientists and Engineers II , Summer Semester 2009
Vector Form of Coulomb’s Law
Force is a vector quantity (has magnitude and direction).
1212221
21221
21 ˆˆ Frr
qqkr
r
qqkF ee
unit vector pointing fromcharge q2 to charge q1
Force exerted by charge q2 on charge q1
(force experienced by charge q1 ).
Physics for Scientists and Engineers II , Summer Semester 2009
Directions of forces and unit vectors
+
+
q1
q212r̂
21F
12F
+
-
q1
q221F
12F
21r̂
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the Resultant Forces on Charge q1 in a Configuration of 3 charges
q3
-
+ +
q1 q2a = 1cm
q1 = q2 = +2.0 C
q3 = - 2.0 C
0.5 cm 0.5 cm
Physics for Scientists and Engineers II , Summer Semester 2009
Forces acting on q1
q3
-
+ +
q1 q2
21F31F
31211 FFF Total force on q1:
Physics for Scientists and Engineers II , Summer Semester 2009
Magnitude of the Various Forces on q1
Nm
C
C
Nm
m
CCkF e
224
212
2
29
221 10596.3100.1
100.41099.8
)010.0(
0.20.2
Nm
C
C
Nm
m
CCkF e
224
212
2
29
231 10192.7105.0
100.41099.8
)25.0010.0(
0.20.2
Note: I am temporarily carrying along extra significant digits in theseintermediate results to avoid rounding errors in the final result.
Physics for Scientists and Engineers II , Summer Semester 2009
Adding the Vectors Using a Coordinate System
q3
-
+ +
q1 q2
21F31F
y
x
Physics for Scientists and Engineers II , Summer Semester 2009
Adding the Vectors Using a Coordinate System
21F31F
y
x
jiFF ˆ0ˆ2121
jFiFF ˆ2
2ˆ2
2313131
jFiFF
FFF
ˆ2
2ˆ2
2313121
31211
Physics for Scientists and Engineers II , Summer Semester 2009
…doing the algebra…
jNiN
jNiNN
jFiFF
FFF
ˆ101.5ˆ105.1
ˆ10086.5ˆ10086.510596.3
ˆ2
2ˆ2
2
22
222
315121
31211
F1 has a magnitude of
NF 21 103.5
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … another example using an even more mathematical approach
q1 = +3.0 C
q2 = - 4.0 C
Charges Location of charges
x1=3.0cm ; y1=2.0cm ; z1=5.0cm
x2=2.0cm ; y2=6.0cm ; z2=2.0cm
In this example, the location of the charges and the distancebetween the charges are harder to visualize Use a more mathematical approach!
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … another example using an even more mathematical approach
12212
21122 r̂
d
qqkFF e
d12=distance between q1 and q2.
.q toq from pointingr unit vecto ˆ 2112 r
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … mathematical approach
We need the distance between the charges. d12 is distance between q1 and q2.
+
x
y
z
1r -
q1
q2
2r
12 rr )()( 121212 rrrrd
Physics for Scientists and Engineers II , Summer Semester 2009
cm
cm
cm
cm
cm
cm
cm
cm
cm
rr
0.3
0.4
0.1
0.5
0.2
0.3
0.2
0.6
0.2
12
cmcmcmcm
cm
cm
cm
cm
cm
cm
rrrrd
260.90.160.1
0.3
0.4
0.1
0.3
0.4
0.1
)()(
222
121212
Distance between charges q1 and q2 .
Calculating the force on q2 … mathematical approach
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … mathematical approach
We need the unit vectors between charges. For example, the unitvector pointing from q1 to q2 is easily obtained by normalizing the vector pointing from from q1 to q2.
+
x
y
z
1r -
q1
q2
2r
12 rr
)()( 121212 rrrrd
12
1212ˆ
d
rrr
12r̂
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … mathematical approach
0.3
0.4
0.1
14
1
0.3
0.4
0.1
26
1ˆ
12
1212
cm
cm
cm
cmd
rrr
The needed unit vector:
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … mathematical approach
126
26
26
9
26
16
26
1ˆ ofLength
263
264
261
263
264
261
12
r
You can easily verify that the length of the unit vector is “1”.
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … another example using an even more mathematical approach
NN
N
m
cm
cm
C
C
Nm
cm
CCk
rd
qqkFF
e
e
2
2
2
212
2
29
2
12212
21122
10
2.1
7.1
42.0
44.124
92.165
48.41
0.3
0.4
0.1
48.41
0.3
0.4
0.1
1
10010462.0109876.8
0.3
0.4
0.1
26
1
26
)0.4(0.3
ˆ
Physics for Scientists and Engineers II , Summer Semester 2009
Calculating the force on q2 … another example using an even more mathematical approach
N
NNN
N
N
N
N
N
N
FFF
2
222
222
101.2
15485275291721
44.124
92.165
48.41
44.124
92.165
48.41
…and if you want to know just the magnitude of the force on q2 :
Physics for Scientists and Engineers II , Summer Semester 2009
.qby divided q charge test aon acting force theas defined is
vector field electric The
:definedFaraday Similarly,
ooe
o
e
FE
q
FE
23.4 The Electric Field
mass.by that divided m mass of particle test aon acting force theas defined is
m. mass theoflocation at the field nalgravitatio The
:2210 Physics fromRemember
g
g
Fg
m
Fg
It is convenient to use positive test charges. Then, the direction ofthe electric force on the test charge is the same as that of the fieldvector. Confusion is avoided.
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field
C
N : field electric of units SI E
again. removed is charge test the
once exists that field actual theis measured field The alone. Q charge source theof field electric the todue
solely is charge test on the force theTherefore, charge. test by the produced field the todue charge testthe
on acting force no is thereHowever, field. electric additionalan produces also charge test thecourse, Of
Q. charge sourceby produced detectsonly charge test The
itself. charge test by the produced NOT is measured The :Note
E
E
+ +
+ ++ +
+ +
+
Source charge
test charge
Q qo E
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field
point.at that placed particle chargedany on force
thecalculatecan you space,in point someat knowyou Once
.directions oppositein are and : q charge negativeFor
. asdirection same in the is : q charge positiveFor
: field electrican in q charge aon Force
E
EqF
EeF
EeF
e
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of a “Point Charge” q
.q toq from pointingr unit vecto theis r̂
and q and qbetween distance theisr where
ˆE:q of place at the qby created field Electric
ˆ:qby q chargeon test exerted Force
0
0
20
0
20
0
rr
qk
q
F
rr
qqkF
ee
ee
q
q0rr̂
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of a Positive “Point Charge” q
+
q0
(Assuming positive test charge q0)
eF
The electric field of a positive point charge points away from it.
+
EP
Force on test charge
Electric field where test chargeused to be (at point P).
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of a Negative “Point Charge” q
-
q0
(Assuming positive test charge q0)
eF
The electric field of a negative point charge points towards it.
-
EP
Force on test charge
Electric field where test chargeused to be (at point P).
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of a Collection of Point Charges
i
i
i
e rr
qk ˆE 2
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
222
212
1
12
ˆˆˆE rr
qkr
r
qkr
r
qk ee
ii
i
e
x
y
P
q1 q2
a b
?E
y
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
2222
1221 ˆˆE r
yb
qkr
ya
qk ee
x
y
P
q1 q2
r2r1
a b
y 2221 yar
2222 ybr
Pythagoras:
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
x
y
P
q1 q2
1̂r 2̂r
1C 2C
R
y
a
yay
a
r
a
yrCR
rr
rr p
2211
1
11
11
11
0
011ˆ
pr1pr1
y
b
yby
b
r
b
yrCR
rr
rr p
2211
2
22
22
11
0
011ˆ
y
b
ybyb
qk
y
a
yaya
qk ee 2222
2
22221 11
E
y
0R
0C1
a
0C2
b
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
y
b
yb
qk
y
a
ya
qk ee
23
22
2
23
22
1E
23
22
2
23
22
1Eyb
bqk
ya
aqk eex
23
22
2
23
22
1Eyb
yqk
ya
yqk eey
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
23
2223
2223
2223
22
2
23
22
1 2E
ya
qak
ya
qak
ya
qak
yb
bqk
ya
aqk eeeeex
0E
23
2223
2223
22
2
23
22
1
ya
qyk
ya
qyk
yb
yqk
ya
yqk eeeey
Special case: q1= q and q2 = -q AND b = a
q -q+ -
E from + charge
E from - charge
component)-y no (has total E
Physics for Scientists and Engineers II , Summer Semester 2009
23.4 The Electric Field of Two Point Charges at Point P
0E
23
2223
2223
22
2
23
22
1
ya
qak
ya
qak
yb
bqk
ya
aqk eeeex
23
2223
2223
2223
22
2
23
22
1 2E
ya
qyk
ya
qyk
ya
qyk
yb
yqk
ya
yqk eeeeey
Special case: q1= q and q2 = q AND b = a
q q+ +
E from + chargeE from other + charge
component)- xno (has total E
Physics for Scientists and Engineers II , Summer Semester 2009
This is called an electric DIPOLE
23
22
2E
ya
qakex
0E y
Special case: q1= q and q2 = -q AND b = a
q -q+ -
E from + charge
E from - charge
component)-y no (has total E
For large distances y (far away from the dipole), y >> a:
32
32
22E
y
qak
y
qak eex E falls off proportional to 1/y3
Fall of faster than field of single charge (only prop. to 1/r2).From a distance the two opposite charges look like they arealmost at the same place and neutralize each other.