Download - Electric Fields
• Electrostatic Phenomena• Coulomb’s Law
12212
2112 ˆ
4
1r
r
qqF
ope
=r
q1• Superposition
qFtotal = F1 + F2 + ...q2
F1
F2
F
Two balls of equal mass are suspended from the ceiling withnonconducting wire. One ball is given a charge +3q and the other is given a charge +q.
g+q+3q
Which of the following best represents the equilibrium positions?
(b)
+3q+q
(a)
+3q+q
(c)
+3q +q
+3q
(b)
+3q+q
(a)
+3q+q
(c)
+q
Which best represents the equilibrium position?
• Remember Newton’s Third Law!• The force on the +3q charge due to the +q charge must be equal
and opposite to the force of the +3q charge on the +q charge• Amount of charge on each ball determines the magnitude of the
force, but each ball experiences the same magnitude of force• Symmetry, therefore, demands (c)
P.S. Knowing the form of Coulomb’s law you can write two equations with two unknowns (T and q )
Two charges q = + 1 μC and Q = +10 μC are placed near each other as shown in the figure.
6) Which of the following diagrams best describes the forces acting on the charges:
Preflight 2:
+10 μC+1 μC
a)
b)
c)
The Electric Field
•A simple, yet profound observation- The net Coulomb force on a given charge is alwaysproportional to the strength of that charge.
q
q1
q2
F1
F
F2
F2F1F = +
÷÷ø
öççè
æ+=
22
222
1
11
0
ˆˆ
4 r
rq
r
rqqF
per
test charge
- We can now define a quantity, the electric field, whichis independent of the test charge, q, and depends only onposition in space:
q
FE
rrº The qi are the sources
of the electric field
The Electric Field
q
FE
rrº
With this concept, we can “map” the electric fieldanywhere in space produced by any arbitrary:
“Net” E at origin
+
F
These charges or this charge distribution“source” the electric field throughout space
ii
i rr
qE ˆ
4
12
0
++
+ +
+
+-
-
--
-
Bunch of Charges
rr
dqE ˆ
4
12
0
+ ++ ++ +++++ +
Charge Distribution
Example: Electric Field
The x and y components of the field at (0,0) are:
1) Notice that the fields from the top-rightand bottom left cancel at the origin?
2) The electric field, then, is just the fieldfrom the top -left charge. It points awayfrom the top-left charge as shown.3) Magnitude of E-field at the origin is:
E2a2
kq=
What is the electric field at the origin due to this set of charges?
x
y
+q+q
+q
a
a
a
a a
2a
Q
Ex 2a2
kq= cosq
2
12a2
kq=
Ey 2a2
kq= sinq-
2
12a2
kq= -
Example: Electric Field
222 a
qQkQEF xx
222 a
qQkQEF yy
Now, a charge, Q, is placed at the origin. What is the net force on that charge?
+q
x
y
+q
+q
a
a
a
a a
2a
Q
Note: If the charge Q is positive, the force will be in the direction of the electric fieldIf the charge Q is negative, the force will be against the direction of the electric field
F is
F is
2
1
2 2a
qkEy
2
1
2 2a
qEx =k
Let’s Try Some Numbers...
2
If q = 5mC, a = 5cm, and Q = 15mC.
Then Ex = 6.364 106 N/C
and Ey = -6.364 106 N/C
Fx= 95.5 N Fy= -95.5 N
Fx = QEx and Fy = QEy
So... and
We also know that the magnitude ofE = 9.00 106 N/C
We can, therefore, calculate the magnitude of F
F = |Q| E = 135 N
x
y
+q+q
+q
a
a
a
a a
2a
Two charges, Q1 and Q2, fixed along the x-axis asshown produce an electric field, E, at a point(x,y)=(0,d) which is directed along the negativey-axis.
Q2Q1 x
y
Ed- Which of the following is true?
(a) Both charges Q1 and Q2 are positive
(b) Both charges Q1 and Q2 are negative
(c) The charges Q1 and Q2 have opposite signs
Two charges, Q1 and Q2, fixed along the x-axis asshown produce an electric field, E, at a point(x,y)=(0,d) which is directed along the negativey-axis.
- Which of the following is true?
E
Q2Q1
(a)
Q2Q1
(c)
E
Q2Q1
(b)
E
Q2Q1 x
y
Ed
(a) Both charges Q1 and Q2 are positive
(b) Both charges Q1 and Q2 are negative
(c) The charges Q1 and Q2 have opposite signs
Ways to Visualize the E FieldConsider the E-field of a positive point charge at the origin
+
+ chg
field lines
+ chg
vector map
+
Rules for Vector Maps
• Direction of arrow indicates direction of field• Length of arrows µ local magnitude of E
+ chg
+
• Lines leave (+) charges and return to (-) charges• Number of lines leaving/entering charge µ amount
of charge• Tangent of line = direction of E• Local density of field lines µ local magnitude of E• Field at two white dots differs by a factor of 4 since r
differs by a factor of 2• Local density of field lines also differs by a factor of 4
(in 3D)
Rules for Field Lines
+ -
3
6) A negative charge is placed in a region of electric fieldas shown in the picture. Which way does it move ?
a) up c) left e) it doesn't move b) down d) right
Preflight 2:
7) Compare the field strengths at points A and B.
a) EA > EB
b) EA = EB
c) EA < EB
• Consider a dipole (2 separated equal and opposite charges) with the y-axis as shown.
+Q
x
a
a-Q
a
2a
y
- Which of the following statements about Ex(2a,a) is true?
(a) Ex(2a,a) < 0 (b) Ex(2a,a) = 0 (c) Ex(2a,a) > 0
Ex Solution: Draw some field lines according to our rules.
• Consider a dipole (2 separated equal and opposite charges) with the y-axis as shown.
+Q
x
a
a-Q
a
2a
y
- Which of the following statements about Ex(2a,a) is true?
(a) Ex(2a,a) < 0 (b) Ex(2a,a) = 0 (c) Ex(2a,a) > 0
Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above.
3) What is the direction of the electric field at point A?
a) up b) down c) left d) right e) zero
4) What is the direction of the electric field at point B?
a) up b) down c) left d) right e) zero
Preflight 2:
Field Lines From Two Opposite ChargesDipole
Dipoles are central to our existence!
Molecular Force Model Basis of Attraction
Ion-dipole Ion and polar molecule
Dipole-dipole Partial charges of polar molecules
London dispersion
Induced dipoles of polarizable molecules
The Electric Dipolesee the appendix for further information
q x
y
a
a
+Q
rE E
-Q
What is the E-field generated bythis arrangement of charges?
Calculate for a point along x-axis: (x, 0)
Ex = ??
Symmetry
Ex(x,0) = 0
Ey = ??
sin4
120,
20 r
QxEy
222 axr r
asin
2/32204
120,
ax
QaxEy
Electric Dipole Field Lines
• Lines leave positive chargeand return to negative charge
What can we observe about E?
• Ex(x,0) = 0
• Ex(0,y) = 0
• Field largest in space between two charges
• We derived:
... for r >> a,
3
1)0,(
xxEy
2/32204
120,
ax
QaxEy
Field Lines From Two Like Charges
• There is a zero halfwaybetween the two charges
• r >> a: looks like the fieldof point charge (+2q) at origin
4
• Consider a circular ring with total charge +Q. The charge is spread uniformly around the ring, as shown, so there is λ = Q/2pR charge per unit length.
• The electric field at the origin is
R
x
y++++
+ +
+++
++
+ + + + ++++++
+
(a) zeroR
2
4
1
0
(b) 204
1
R
R
(c)
• Consider a circular ring with total charge +Q. The charge is spread uniformly around the ring, as shown, so there is λ = Q/2pR charge per unit length.
• The electric field at the origin is
R
x
y++++
+ +
+++
++
+ + + + ++++++
+
(a) zeroR
2
4
1
0
(b) 204
1
R
R
(c)
• The key thing to remember here is that the total field at the origin is given by the VECTOR SUM of the contributions from all bits of charge.
• If the total field were given by the ALGEBRAIC SUM, then (b) would be correct. (exercise for the student).
• Note that the electric field at the origin produced by one bit of charge is exactly cancelled by that produced by the bit of charge diametrically opposite!!
• Therefore, the VECTOR SUM of all these contributions is ZERO!!
Electric Field inside a Conductor
• A two electron atom, e.g., Ca– heavy ion core– two valence electrons
• An array of these atoms– microscopically crystalline– ions are immobile– electrons can move easily
• Viewed macroscopically:– neutral
There is never a net electric field inside a conductor – the free charges always
move to exactly cancel it out.
2+
2+ 2+ 2+ 2+
2+ 2+ 2+ 2+
2+ 2+ 2+ 2+
2+ 2+ 2+ 2+
Summary
• Define E-Field in terms of forceon “test charge”
• How to think about fields
• Electric Field Lines
• Example Calculation: Electric Dipole
Appendix A:Other ways to Visualize the E Field
Consider a point charge at the origin
+
+ chg
Field LinesEx, Ey, Ez as a function of (x, y, z)Er, Eq, Ef as a function of (r, q, f)
Graphs
x
Ex(x,0,0)
Appendix A- “ACT”
(a) (c)(b)
f0 2p
Er3A
f0 2p
Er
f0 2p
Er
Fixedr > 0
3B
Fixedr > 0
f0 2p
Ex
f0 2p
Ex
f0 2p
Ex
Consider a point charge fixed at the origin ofa coordinate system as shown.
–Which of the following graphs bestrepresent the functional dependence ofthe Electric Field for fixed radius r?
xQ
y
r
f
Appendix A “ACT”Consider a point charge fixed at the origin ofa coordinate system as shown.
– Which of the following graphs bestrepresent the functional dependence ofthe Electric Field for fixed radius r?
xQ
y
r
f
(a) (c)(b)
f0 2p
Er3A
f0 2p
Er
f0 2p
Er
Fixedr > 0
• At fixed r, the radial component of the field is a constant, independent of f!!
• For r>0, this constant is > 0. (note: the azimuthal component Ef is, however, zero)
Appendix A “ACT” Consider a point charge fixed at the origin ofa coordinate system as shown.
–Which of the following graphs bestrepresent the functional dependence ofthe Electric Field for fixed radius r?
xQ
y
r
f
(a) (c)(b)
3B
Fixedr > 0
f0 2p
Ex
f0 2p
Ex
f0 2p
Ex
cosrx EE • At fixed r, the horizontal component of the field Ex is given by:
Appendix B: Electric Dipole
What is the E-field generated bythis arrangement of charges?
Calculate for a point along x-axis: (x, 0)
Ex = ??
Symmetry
Ex(x,0) = 0
Ey = ??
sin4
120,
20 r
QxEy
222 axr r
asin
2/32204
120,
ax
QaxEy
q x
y
a
a
+Q
rE E
-Q
Electric Dipole
x
y
a
a
+Q
-Q
0,0 yEx
Coulomb ForceRadial
22
0
11
4,0
ayay
QyEy
2
2
240
1
4
4,0
ya
y
ayQyEy
E
Now calculate for a pt along y-axis: (0,y)
Ex = ?? Ey = ??
What is the Electric Field generated by this charge arrangement?
Electric Dipole
x
y
a
+Q
-Q
Case of special interest:(antennas, molecules)
r > > a
r
For pts along x-axis: For pts along y-axis:
2/32204
120,
ar
QarEy
00, rEx
3
04
120,
r
QarEy
For r >>a,
E rx 0 0,
2
2
240
1
4
4,0
ra
r
arQrEy
For r >>a
3
04
14,0
r
QarEy
a