Transcript Electric Field - Department of Physics

Electric Field
Point charge:
q1
E=
r̂
2
4pe 0 r
Esphere
1
Dipole:
for r>>s :
s
+q
z
Q
=
rˆ
2
4pe 0 r
1
Esphere = 0
y
-q
Uniformly charged sphere:
x
Dipole moment: p = qs
for r>R (outside)
for r<R (inside)
1 2qs
E=
,0,0
3
4pe 0 r
at <r,0,0>
-1 qs
E=
,0,0
3
4pe 0 r
at <0,r,0>
-1 qs
E=
,0,0
3
4pe 0 r
at <0,0,r>
Clicker Question 1
What is the magnitude of the net electric field at
location X, due to these two charges? Assume d >> s
s
-e
+2e
Choice
A
B
C
𝑬𝒏𝒆𝒕
1 𝑒𝑠
4𝜋𝜀0 𝑑 3
1 2𝑒𝑠
4𝜋𝜀0 𝑑3
𝑛𝑜𝑛𝑒
𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
d
X
What is the approximate
magnitude?
Choice
A
B
C
𝑬𝒏𝒆𝒕
1 𝑒
4𝜋𝜀0 𝑑 2
1 2𝑒𝑠
4𝜋𝜀0 𝑑 3
1 1.5𝑒𝑠
4𝜋𝜀0 𝑑 3
Clicker Question 2
Locations A, B, and C are equidistant from the center of
the dipole (charges +q and –q are separated by s). At
which location(s) is the magnitude of the electric field
𝟏 𝒒𝒔
approximately
?
𝟑
𝟒𝝅𝜺𝟎 𝒅
d
+
A
A. at location A
d
B. at location B
d
C. at location C
B
D. at locations A, C
E. at locations A, B, C
C
Clicker Question 3
What is the direction of the electric field at location X,
due to the dipole?
+
C
B
D
A
E
X
Clicker Question
Clicker Question
Clicker Question
Choice of System
Multiparticle systems: Split into objects to include into system
and objects to be considered as external.
To use field concept instead of Coulomb’s law we split the
Universe into two parts:
• the charges that are the sources of the field
• the charge that is affected by that field
A Fundamental Rationale
• Convenience: know E at some
r location – know the electric
force on any charge: F  qE
• Electric properties of matter– independent of how this
field was produced.
Example: if E > 3106 N/C air becomes a conductor
• Retardation
Nothing can move faster than light c
c = 300,000 km/s = 30 cm/ns
Coulomb’s law is not completely correct – it does not
contain time t nor speed of light c.

F
1 q1q2
rˆ
2
40 r

E
1
q
rˆ
2
40 r
v<<c !!!
Chapter 15
Matter and Electric
Fields
Net Charge
Matter is made out of atoms.
Atom contains charged particles: electrons (-e), protons (+e)
Neutral atom: number of electrons and protons is equal:
Example: Hydrogen atom: 1 proton, 1 electron
net charge = (+e) + (-e)=0
Sodium atom: 11 protons, 11 electrons
Sodium atom (Na) can lose an electron:
Sodium ion (Na+): (+11e) + (-10e) = +e
Ordinary matter is electrically neutral.
However, can be charged by adding/removing charged particles
Conservation of Charge
The net charge of a system and
its surroundings cannot change
If one object gets charged positively, there must be an object
which gets charged negatively.
The net electric charge is conserved in any physical process.
Charge can be transferred from one object to another.
Pair-Production: 𝛾 → 𝑒 + + 𝑒 −
The Structure of an Atom
Hydrogen
10-10 m (1 Å)
Nucleus, ~10-15 m
Charge of electron cloud equals that of nucleus  neutral atom.
If the electron cloud is centered on the nucleus  electric field
produced by electrons exactly cancels the field produced by
nucleus.
Polarization of Atoms
E
+
-
+
Force due to E created by positive charge shifts electron
cloud and nucleus in opposite directions: electric dipole.
An atom is said to be polarized when its electron cloud has
been shifted by the influence of an external charge so that
the electron cloud is not centered on the nucleus.
Induced Dipole
An applied electric field creates induced dipoles!
E
• it is not a permanent dipole
• an induced dipole is created when a neutral object is polarized
by an applied electric field
Polarization
Amount of polarization p in most materials is proportional to the
magnitude of the applied electric field:
p = aE
𝛼 - “polarizability” of a material
In an induced dipole, is the distance between the charges fixed?
The distance is proportional to the strength of the applied field.
+
−+
E
− +
E
A Neutral Atom and a Point Charge
1. Charge q1 creates field E1 at the location of the atom
q1
E1 =
rˆ
2
4pe 0 r
1
A Neutral Atom and a Point Charge
1) E1 =
q1
rˆ
2
4pe 0 r
1
2. Field E1 polarizes the atom creating a dipole
1 aq1
p = aE1 =
rˆ
2
4pe 0 r
A Neutral Atom and a Point Charge
1) E1 =
q1
rˆ
2
4pe 0 r
2) p =
1 aq1
rˆ
2
4pe 0 r
1
3. Dipole creates field E2 at the location of q1
æ 1 ö 2aq1
1 2p
1 2a
E2 =
=
E1 = ç
r̂
÷
3
3
5
4 pe 0 r
4 pe 0 r
è 4 pe 0 ø r
2
A Neutral Atom and a Point Charge
q1
E
=
1) 1 4pe r 2 rˆ
0
2
1
2) p =
3)
1 aq1
rˆ
2
4pe 0 r
4. Induced dipole exerts force F1 on the charge:
2
æ 1 ö 2aq12
÷÷
F1 = q1E2 = çç
rˆ
5
è 4pe 0 ø r
æ 1 ö 2aq1
÷÷
E2 = çç
rˆ
5
è 4pe 0 ø r
A Neutral Atom and a Point Charge
q1
E
=
1) 1 4pe r 2 rˆ
0
2
1
1 aq1
p
=
rˆ
2)
2
4pe 0 r
3)
æ 1 ö 2aq1
÷÷
E2 = çç
rˆ
5
è 4pe 0 ø r
4)
æ 1 ö 2aq12
÷÷
F1 = çç
rˆ
5
è 4pe 0 ø r
2
5. The charge q1 exerts force F2 on the dipole (reciprocity):
2
æ 1 ö 2aq12
÷÷
F2 = - F1 = -çç
rˆ
5
è 4pe 0 ø r
A Neutral Atom and a Point Charge
1) E1 =
q1
rˆ
2
4pe 0 r
2) p =
1 aq1
rˆ
2
4pe 0 r
1
2
3)
æ 1 ö 2aq1
÷÷
E2 = çç
rˆ
5
è 4pe 0 ø r
4)
æ 1 ö 2aq12
÷÷
F1 = çç
rˆ
5
è 4pe 0 ø r
5)
æ 1 ö 2aq12
÷÷
F2 = -çç
rˆ
5
è 4pe 0 ø r
2
2
Neutral atoms are attracted by charges!
Interaction strength ~ 1/r5
Exercise
Atom A is easier to polarize than atom B. Which atom would
experience a greater attraction to a point charge a distance r away?
A
B
-
FA
+
-
+
FB
2
æ 1 ö 2aq12
÷÷
F2 = çç
~a
5
è 4pe 0 ø r
Interaction of Charged Tapes and Neutral
Matter
Interaction of like-charged Objects
q1
q2
FR
Conductor
Repulsion:
FA
Plastic
1 q1q2
FR =
rˆ
2
4pe 0 r
2
Attraction:
Total:
æ 1 ö 2aq12
÷÷
FA = - N çç
rˆ
5
è 4pe 0 ø r
2
2ù
é 1 qq
æ
ö
1
2
a
q
1 2
1
ç
÷
F=ê
N
ú rˆ
2
5
ç
÷
êë 4pe 0 r
è 4pe 0 ø r úû
Determining the Charge of an Object
Suppose tape is negatively charged, and you
rub a wooden pencil on a wool sweater and
bring it near the tape.
If tape swings toward the pencil, does it
show that the pencil had been charged
positively?

--
NOT NECESSARILY!
Attraction: can happen for like-charged objects!
Repulsion: can happen only for like-charged objects!
Electric Field Through Intervening Matter
F
F
The field appears to be weaker in presence of intervening
(polarizable) object. Both tapes are attracted to paper
Superposition principle: the presence of matter does not affect
the electric field produced by a charged object.
Intervening matter does not “block” the E field
The resulting field is a superposition of two fields:
Field of the other charge plus the field of induced dipoles.