Lecture 2 - Coulomb`s Law and Electric Field

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Lecture 2 - Coulomb’s Law and Electric Field
Chapters 25 and 26 - Thursday January 11th
•Quick review of Tuesday’s class
•Coulomb’s Law
•Scalar and vector notation
•Discrete charge distributions (superposition principle)
•Continuous charge distributions
•Electric fields
•Discrete charge distributions
•Continuous charge distributions
•Electric field lines
Reading: pages 567 thru 597 (chapters 25 & 26) in HRK
Read and understand the sample problems
WebAssign: set 1, due Thur. 18th at 11:59pm
Graded problems: Ch. 26 – Ex. 8, 13, 18, 36; Prob. 10
Practice problems: Ch. 26 - Ex. 7, 14, 23 ; Prob. 1
What is charge?
•Charge is measured in Coulomb’s (C)
•Fundamental unit.
•Definition based on forces between current carrying
wires (current = Ampères, or C/s), i.e. chapter 33.
•Charge is discrete
•Thompson discovered the electron in 1896. He found
that charge was carried by elementary particles with
the same charge to mass ratio.
•The elementary charge of the electron was not
measured until 1909 (Millikan).
•Both experiments earned Nobel prizes.
Charge on an electron:
1 Coulomb of charge:
1 Ampère (= 1 C/s)
Charge is discrete:
e = 1.6 × 10−19 Coulombs
6.24 × 1018 electrons
6.24 × 1018 electrons/second
q = ne
n = ±1, ±2, ±3,...
Static electricity through charging
Conductor
Insulator
Coulomb’s Law
Coulomb’s torsional balance
1785
q1 q2
F∝
,
2
r
K=
1
4πε o
or
q1 q2
F=K
r2
= 8.99 × 109 N ⋅ m 2 / C2
εο = 8.85418781762 × 10-12 C2/N·m2
1
εo =
−7
2
2
2
4
10
N
s
/
C
π
×
⋅
×
c
(
)
c = speed of light in vacuum
= 299792458 m/s
Coulomb’s Law in vector notation
G
F12 =
G
1 q1q2
rˆ12 = − F21
2
4πε o r12
G
F21 =
1 q1q2
rˆ21
2
4πε o r21
G
r12
rˆ12 =
= −rˆ21
r12
Superposition principle
Source
charges
Test charge
q0
G
G G G
G
G
Fq0 = F1 + F2 + F3 + ...Fi = ∑ Fi
i
•Leads to Maxwell’s equations being linear.
Calculus (continuous charge distribution):
G
G
F = ∫ dF = iˆ ∫ dFx + ˆj ∫ dFy + kˆ ∫ dFz
G
dF =
q0 dq
rˆ
2
4πε o r
1
q0 = test charge
Charge densities
In 1D (a line or wire):
Q
λ= ,
L
or
dQ
λ=
dL
λ is the line charge density, or charge per unit length, in Coulombs
per meter. L represents length, and Q is charge.
In 2D (a surface or sheet): σ = Q ,
A
or
dQ
σ=
dA
σ is the surface charge density, or charge per unit area in Coulombs
per meter2; A represents area, and Q is charge.
In 3D (a solid object):
Q
ρ= ,
V
or
ρ is the volume charge density, or charge per unit volume in
Coulombs per meter3. V represents volume, and Q is charge.
dQ
ρ=
dV
Electric field - Chapter 26
G
GMm
m
Analogy with gravitation
F = − 2 rˆ
r
•Problem:
Force depends on test mass!
•But:
G
G
G
G F
F = mg or g =
m
G
GM
g = − 2 rˆ
r
• Same for all masses.
• Represents the gravitational
influence (field) of the mass M.
Electric field - Chapter 26
G
qq
q0 +
Analogy with gravitation
F = K 20 rˆ
r
•Problem:
Force depends on test charge!
G
G
G
G F
F = q0E or E =
q0
•Definition:
G
E=
1
q
rˆ
2
4πε o r
• Same for all test charges.
• Represents the electrostatic
influence (field) of the charge q.
source charge
q = −Q
Electric field - Chapter 26
Newton’s law for electrostatics:
G
G
F = q0E
There’s really no need for the “test charge”
G
G
F = qE
G
This is the force on a charge q in an electric field E
Units for E are N/C in this chapter
(later we shall use volts per meter)
Electric field - Chapter 26
Coulomb’s law:
G
E=
1
q
ˆ
r
2
4πε o r
•Note that the force may be directed both towards and
away from the source charge, since this charge may be
either positive or negative.
G
•Length of arrow signifies magnitude of E
Of course, superposition principle still holds:
G G G
G
G
G
E = E1 + E2 + E3 + ...Ei = ∑ Ei
i
G
G
E = ∫ dE = iˆ ∫ dE x + ˆj ∫ dE y + kˆ ∫ dE z
Electric field lines
• The tangent to an electric field line at a point in space
gives the direction of the electric field at that point.
• The magnitude of the electric field at any point is
proportional to the number of field lines per unit crosssectional area perpendicular to the lines.
Electric field lines
• Electric field lines start on positive charges and end on
negative charges (can also start/end at infinity).
• The symmetry of the problem dictates the directions in
which field lines radiate from charges.
Electric field lines
• The number of field lines radiating from a charge is
proportional to the charge.
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