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Physics 227: Lecture 2
Coulomb’s Law, Superposition,
Electric Fields, Field Lines
Lecture 1 review:
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All your questions are answered on class web pages: Sakai:
01:750:227 or http://www.physics.rutgers.edu/ugrad/227
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Separate charges by rubbing (appropriate materials).
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There are two types of charges: + and -.
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Opposites signs attract, same sign repel.
Conductors conduct, insulators do not.
Charged objects ``polarize’’ uncharged insulators or
conductors, leading to an attractive force.
Friday, September 9, 2011
Coulomb’s Law - force between two charges
Consider two point charges q1
and q2 which have the same
sign. The force is along the
line connecting the charges.
F2 on 1 = k q1 q2 / r2
q1
q2
F1 on 2 = k q1 q2 / r2
If q1, q2 have opposite signs:
k = 8.99x109 N.m2/C2
or k = 1/4πε0 with
ε0 = 8.854x10-12 C2/N.m2
permittivity of free space
q1
q2
F1 on 2 = k q1 q2 / r2
F2 on 1 = k q1 q2 / r2
Note the force is negative
for charges of opposite
signs - this indicates the
force is attractive.
Note: until further notice, we are dealing with point charges, or with charge
distributions that we assume are not significantly affected by the presence of other
external charges. We ignore polarization effects such as we saw between a charged
rod and uncharged insulators or conductors in the first lecture.
Friday, September 9, 2011
Coulomb’s Law - force between two charges
Consider two point charges q1
and q2 which have the same
sign. The force is along the
line connecting the charges.
q1
q2
More formally, with vectors, Coulomb’s law is:
In using this definition, you need to recall the definition of the
direction of the unit vector as going from q1 to q2. Thus the force of q1
on q2 is in the direction from q1 to q2, if q1 and q2 have the same sign.
Friday, September 9, 2011
Coulomb vs. Gravitational Force
Compare the strength of the Coulomb
force to the strength of the gravitational
force, between an electron and proton:
|FC| = k |q1 q2| / r2 = 9.0x109 n.m2/C2 (1.6x10-19 C)2 = 2.3x10-28 N / r2
|FG| = G m1 m2 / r2 = 6.7x10-11 n.m2/C2 (1.7x10-27 kg) (9.1x10-31 kg) =
1.0x10-67 N / r2
Both fall as 1/r2, so the distance does not matter.
FC / FG ≈ 2x1039
Note: mearth ≈ 6.0x1024 kg so earth has ≈ 3.6x1051 protons + neutrons.
Friday, September 9, 2011
Superposition
The total force on an object is the vector
sum of all the forces on the object.
F2 on 1 = k q1 q2 / r2
q3
q1
q2
F3 on 1 = k q1 q3 / r2
FTotal on 1
Recall from introductory mechanics / statics that
the sum of the internal forces within an object is
0, so we will not worry about those forces here.
Friday, September 9, 2011
Reminder - Vector Addition
A
Graphical addition:
Bslid
C
B
Slide tail of B to head of A, C
goes from tail of A to head of B
Recall that a vector has a magnitude and direction. Its
absolute position does not matter. If you slide it around, it is
still the same vector.
Algebraic addition (2-dimensional example):
Friday, September 9, 2011
Simple Superposition Example
q1
q2
2C
-1 C
x=-1 m, y=0
x=0, y=0
q3
1C
x=1 m, y=0
What is the total Coulomb force on the -1 C charge at the origin?
Friday, September 9, 2011
Iclicker:
Superposition:
x=0, y=1 m
1C
-1 C
x=0, y=0
What is the total Coulomb force
on the -1 C charge at the origin
from the other two 1 C charges?
1C
x=1 m, y=0
A. k
B. k
C. k
D. k
E. k
Friday, September 9, 2011
Iclicker:
Superposition:
x=0, y=1 m
1C
-1 C
x=0, y=0
What is the total Coulomb force
on the -1 C charge at the origin
from the other two 1 C charges?
1C
x=1 m, y=0
A. k
B. k
C. k
D. k
E. k
Friday, September 9, 2011
Electric Field
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We introduce the electric field from the force felt by a
``test’’ charge:
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Alternately, the electric field is the force felt by a unit
charge, divided by 1 C (to get the units right).
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In doing this, we assume that the charges generating the
force are fixed in place - they do not move around in
response to the test / unit charge.
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The electric field should remind you of the gravitational
field. For point masses / charges:
�C
F�g
GMe
F
kQ
Q
� =
�g =
= − 2 r̂ → E
= 2 r̂ =
r̂
2
m
r
q
r
4π�0 r
E is a ``vector field’’ - its magnitude and direction depend on position.
Friday, September 9, 2011
A note on Electric Fields
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For now it might just seem that the concept of electric
fields is some math trick that does not make much
difference - the electric field is some math abstraction
rather than something that is real.
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We will later learn that the fields are real. For example:
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Energy is stored in electric and magnetic fields.
Light is a traveling electric + magnetic field.
Recall Einstein’s conception of gravity (General Relativity)
geometrically, as curving space rather than being a force.
Similarly, you can think a charge generates an electric
field, like a mass generates a gravitational field, modifying
space.
Friday, September 9, 2011
Superposition of Electric Fields
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Since forces are vectors, and add as vectors
And FC = qE...
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➭ Electric fields are vectors and add as vectors
If we have charges qa, qb, qc, qd, qe, ... the force on charge
qa is:
When we are
finding the force
on a charge qa, we
only consider
Or Ftotal = qaE, with the electric fields adding:
fields generated
by the other
charges, not by qa.
Friday, September 9, 2011
Field Lines
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Field lines show the electric field direction. By convention, field
lines point from + to - charges. Field lines start on + charges
and end on negative charges - or can start / stop at r = ∞.
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The density of field lines indicates the magnitude of the field.
The field gets smaller away from a charge. A larger charge
has more field lines that start / stop at it.
q2>0
Friday, September 9, 2011
q1<0
Field Lines for Two Charges
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In the middle and right drawings, the two charges have the same
number of field lines, so are equal in magnitude.
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two charges, the field lines curve. The E field at each point is
tangent to the field line at that point - field lines cannot cross!
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In the middle drawing, all the field lines start on the + charge and
curve around to end on the - charge, except for two of the three
horizontal ones which reach to r = ∞.
Friday, September 9, 2011
Field Lines for Two Charges
• Why
don’t we draw horizontal
field lines pointing from each
positive charge to the other?
A. We have enough lines
already.
B. The fields from the two
charges cancel between
them.
C. The lines would point at
each other.
D. The lines would not end at a
charge or at r=∞.
E. Blue lines are not pretty. We
need Rutgers red!
Friday, September 9, 2011
Field Lines for Two Charges
• Why
don’t we draw horizontal
field lines pointing from each
positive charge to the other?
A. We have enough lines
already.
B. The fields from the two
charges cancel between
them.
C. The lines would point at
each other.
D. The lines would not end at a
charge or at r=∞.
Field lines start/stop on charge or at r =
E. Blue lines are not pretty. We
∞, have a unique magnitude & direction at
need Rutgers red!
each point in space, do not touch or cross
Friday, September 9, 2011
iClicker: Field Lines for Three Charges
What is the most you
can determine about
qtop, qmiddle and qbottom?
A. qt > qm > qb
B. qt > qm < qb
C. qt, qb > 0, qm < 0
D. qt, qb > 0, qm < 0
|qt| = |qb| > |qm|,
E. |qt| = |qb| > |qm|,
qt, qb < 0, qm > 0
Friday, September 9, 2011
iClicker: Field Lines for Three Charges
What is the most you
can determine about
qtop, qmiddle and qbottom?
A. qt > qm > qb
B. qt > qm < qb
C. qt, qb > 0, qm < 0
D. qt, qb > 0, qm < 0
|qt| = |qb| > |qm|,
E. |qt| = |qb| > |qm|,
qt, qb < 0, qm > 0
Friday, September 9, 2011
Thank you, and
See you next Monday
Friday, September 9, 2011