Download Chapter 17 - Helmut Katzgraber

What is the scope of the class?
Physics 202
Helmut G. Katzgraber
Scope:
Electricity and magnetism
Light and optical instruments
Quantum effects due to light
Particle–wave duality
Atomic structures
Radioactivity and nuclear physics.
Why should anyone care? After all, it’s all boring physics!
Understanding the underlying physical processes can help a lot.
Knowing a bit about E&M and circuitry is useful for lab work.
I guess I do not need to elaborate on the radioactive part...
FS 10
Chapter 17: Electric charge & field
What will we learn in this chapter?
Interactions between electric charges
Coulomb’s law
Electric charges are quantized, i.e., the total charge is a multiple of e
Electric charges obey a conservation law
Notion of a “field”
Have we felt charges before? Yes!
Scuffing feet
Exiting car
Combing hair
Wiping certain materials
...
How many charges exist?
Did we “create” charges?
No.
The experiment shows that charge was transferred from the fur/
silk onto/off the plastic/glass.
We cannot create electric charge, but we can move negativelycharged electrons from one object to the other leaving positivelycharged “holes” behind.
The total electric charge of both cloth and bar does not change.
This is a simple example of a conservation law.
fur – plastic
silk – glass
rubbed plastic – rubbed glass
The simple experiment shows that there are only two types of
charges: postivie (+) and negative (–).
Same charges repel (+ + and – –), opposite attract (+ –).
Physical basis of electric charge
Neat experiment:
Take a transparent balloon and add some tiny styrofoam pellets.
Inflate and rub the balloon. The pellets will charge and form a
mesh in space.
Physical basis of electric charge contd.
The interactions responsible for the structure and stability of atoms,
molecules and matter are mainly electrical interactions between
charged particles.
nucleus
100000 times
smaller
(~2000 lighter than protons)
Note: The charge of the proton and
electron are equal.
Net charge of a neutral atom is zero.
Number of protons: Atomic number.
The process of having nonzero charge is called ionization.
Nuclei are not free to move in solids (charge due to electrons).
Conductors vs insulators
Conductors vs insulators contd.
Conductors:
Metals (outer electrons can move freely trough the lattice).
Ionic liquids (e.g., NaCl+H20) and charged gases (charges can
move freely).
Note: Ionic solutions play a crucial role in biology (e.g., neurons).
Insulators:
Non-metals (electrons are bound to the nucleus).
Conductors: permit charges to move.
Insulators: do not transfer charges.
Oddballs:
Semiconductors (between conductors and insulators).
Superconductors (perfect conductors – no resistivity).
In general, good insulators can be
“charged,” whereas conductors
“spread” the charge.
Induction
Idea: charge an object
without losing any charge.
One can imagine “positive”
charges (holes) moving.
A electron deficiency
corresponds to a positive
charge.
Quick review of some concepts….
e- buildup
If you have a charged ball, where are the charges located?
Evenly distributed?
Center of the ball?
Surface?
Keep in mind that same charges repel each other!
3 identical balls are mounted on insulators. Ball A has a charge Q,
balles B and C are uncharged. Ball A touches ball B, and then touches
ball C. What is the charge on ball A?
Q/2
Q/3
Q/4
!
Understanding air purifiers (not a scam)
Charged objects can exert forces on neutral things:
Balloons stick to walls
Paper sticks to comb
But first a comb:
The charge on the comb attracts the
electrons in the paper, inducing a
net charge.
Induced charge effect,
the system polarizes.
Air purifiers:
Charged plates
make dust and
pollen stick.
Conductors: electrons
move around the system.
Insulators: tiny shifts of
electron clouds around
atoms (adds up!).
Conservation & quantization of charge
In a closed system:
The algebraic sum of all electric charges in any closed
system is constant. Charge can be transferred from one
object to another, and that is the only way that an object
can acquire net charge.
Notes:
This is a universal conservation law.
Even in colliders, where particles are
created/destroyed, charge is conserved.
LHC
The magnitude of the electron/proton
charge is a natural unit. Any amount of observable charge is a
multiple of it (there are no half cents).
Electric forces hold atoms and molecules together.
Other applications:
Car paint.
Laser printers.
...
Coulomb’s law
Coulomb studied forces between charged objects with a
torsion balance.
Result: The force depends
! ! only on the distance
! ! and the amount of
! ! charge.
C. A. de Coulomb
(1736-1806)
The magnitude F of the force between two point
charges q1 and q2 is directly proportional to their
product and inversely proportional to the square
of their distance:
1 q1 q2
�0 = 8.854 × 10−12 C2 /(Nm2 )
F =
4π�0 r2
force measurement due to
torsion of the wire
Polarization
Coulomb’s law contd.
How many Newtons for one Coulomb?
Notes:
Coulomb forces obey
Newton’s third law.
Forces between 2 point
charges act along a line
and are equal in magnitude.
Coulomb’s law is strictly
valid for point charges in
vacuum only.
The principle of superposition
for more than one charge holds:
When adding several forces, you
need to use vector sums.
Unit of charge:
Coulomb. The charge of an electron is e = 1.602 × 10−19 C
Electric fields and forces
Remember:
1 q1 q2
4π�0 r2
e = 1.602 × 10−19 C
F =
What does this all mean?
Two balls with 1C charge at 1m distance repel with 9 109 N:
1
9
k=
= 8.9875 109 Nm2 /C2 → F ≈ 8.98 10 N
4πε0
Typically we encounter values of 10-9 to 10-6C.
One gram of material has approximately 1023 particles.
This means we have potentially 104 – 105C.
Electrical neutrality can only be disturbed with huge
forces.
Definition: electric field
Remove charge q’ and replace
by point P.
We say that A causes an
electric field E at point P.
When a “test charge” is placed
at P, it feels a force F’.
Since P can move, the electric
field E exists around A.
When a charged particle with charge q’ at point P is acted upon by an
� at that
electric force F� , the electric field E
�
� =F
point is defined as
E
q�
If q’ is positive, the force and field point in the same direction; if
negative in opposite directions.!! “Propensity to generate a force”
q� > 0
To see if there is a field
at point P, we place a
test charge at P and
measure the force.
�
Note: F� → E
q� < 0
Units (SI): 1 N/C (Newton/Coulomb)
Note: the force acting on q’ depends on its location in space. Thus
the electric field is not a simple vector, but a vector field; i.e., a vector
�
associated with every point in space. Thus each component of E
�
depends on spatial coordinates, i.e., E = (Ex (�r), Ey (�r), Ez (�r)).
Calculating electric fields
Principle of superposition: The total electric field at any point due to
two or more charges is the vector sum of the fields that would be
produced at that point by the individual charges.
�1
S
Example: 3 charges at source points S.
q1
�3
What is the field at field point P?
S
�2
S
�
q3
Compute individual fields Ei acting on
q2
a test charge at P.
P
� =E
�1 + E
�2 + E
�3
The total field at P is then: E
� at point P due to a point
The magnitude of the electric field E = |E|
charge q at S at a distance r from P is
1 |q|
E=
4π�0 r2
By definition, if q > 0 the field points away from the charge, if q < 0
points towards the point charge.
Electric field lines
In general, only animals can feel electric fields (e.g., birds).
Visualizing electric fields:
Electric field lines are imaginary lines for which the tangent at
each point is the electric field.
� at each point.
Field lines show the direction of E
The spacing between the lines gives an idea of the magnitude.
The electric field is unique, hence only one field line passes
through one point: lines never cross.
Convention: field lines point away from positive charges.
weak
strong
Special case: spherical point charges
For spherically-symmetric charge
distributions (e.g., an insulating
sphere) the produced field
outside the distribution is the
same as if the charge were
concentrated in P.
To obtain the field outside
a sphere, one can assume it
collapsed to a point P with
the total charge of the
sphere Q.
This is useful when computing
fields between e.g., spheres and
walls.
P
Examples of typical field patterns
into negative charges
positive point charge
tangent to the field
dipole
close when strong field
two positive point charges
Note:
These are cross-sections of the 3D
patterns.
Uniform field (needed in applications):
parallel plate capacitor. Inside, the field
is nearly uniform.
Gauss’s law
Historical interlude: Carl Friedrich Gauss
Equivalent to Coulomb’s law.
Provides an alternative approach to calculating electric fields.
Allows us to find the field at point P caused by a single point charge q.
How do we deal with extended charge distributions?
We represent them as sums (integrals) over point charges (use the
superposition principle here).
What is the concept behind Gauss’s law?
Surround a charge distribution with an imaginary closed surface.
Study the electric field at various points on the surface.
Gauss’s law relates the field on all points on the surface and the
total charge enclosed.
Electric flux
C. F. Gauss
(1777 - 1855)
Some things you will
encounter:
Gauss distribution
Gauss’s law
Gauss’s law
Think of liquid flowing trough a surface.
Definition:
�.
Consider a small area A perpendicular to E
The electric flux is then given by ΦE = EA .
If the area is not perpendicular, we only consider the
� , E⊥ = E cos(φ) : ΦE = EA cos(φ)
perpendicular component of E
ΦE = EA
German mathematician.
Possibly one of the most influential ever.
Contributions in statistics, calculus, linear
geometry, geodesy, electrostatics, astronomy, ….
Did not like to publish his work. Had he published
all his results at once, he would have advanced
science back then by approx. 50 years! �
100
x
Child prodigy: in primary school added
in second.
x=1
ΦE = EA cos(φ)
ΦE = EA cos(90◦ ) = 0
The total electric flux ΦE coming out of a closed surface is
proportional to the total electric charge Qencl inside the surface,
according to the relation �
E⊥ ∆A = 4πkQencl = Qencl /�0
The sum represents the operation of dividing a surface A into small
elements ∆A and then computing the small flux contributions.
k = 1/4π�0
Note:
If Qencl = 0 the total flux must be zero.�
� A
� = Qencl /�0.
Ed
In the continuum limit, we would write
S
Gauss’s law is the “integral form” of one of Maxwell’s equations.
These 4 equations can be used to derive all E&M equations you
will learn in this course!
Derivation of Gauss’s law for a point charge
Assume a single positive point charge q.
Place it at the center of an imaginary sphere of
radius R.
The magnitude of the electric field E at every
point of the surface is given by (why?)
q
E=k 2
k = 1/4π�0
R
� is always perpendicular to the surface with total
On the sphere E
2
area A = 4πR . Thus we obtain for the flux
q
ΦE = EA = k 2 · 4πR2 = 4πkq (spherical surface)
R
The flux is independent of R, it only depends on the charge q!
A generalization to arbitrary surfaces follows by dividing the surface
into small elements ∆A.
Faraday ice pail
Experiment to prove with high precision Gauss’s / Coulomb’s laws.
Place charged ball inside conducting bucket.
Close the lid; charges are induced in the walls of the container.
Let the ball touch the bucket. The ball becomes part of the cavity
surface. If Gauss’s law is correct, the charge of the inner surface
(bucket+ball) must be zero. The ball must lose all its charge.
Pull out the ball and measure. The charge is zero.
Charges on conductors
So far we have learned:
In electrostatic situations (no charge motion) the electric field at
every point within a conductor is zero (otherwise the charges
would move).
In a solid conductor, charges distribute on the surface.
solid conductor
with charge q’
solid conductor with
charge q’ with a cavity
an isolated charge q is
placed inside the cavity
Gaussian
surface
charges are entirely
on the surface
� = 0 inside
E
because the field is zero within
the conductor, the field on the
Gaussian surface must be zero.
there cannot be charge on the
cavity surface
for the field to be zero at all
points on the Gaussian
surface, the cavity surface
must have a net charge –q.
Application: Electromagnetic shielding
Why:
To protect sensitive instruments.
To prevent students from using
their cellphones.
Protects people from lightning
when inside cavities (planes, cars).
(Faraday cage)
How?
Surround the object to be
protected with conducting
material.
The field redistributes the charges in the conductor, causing
another electric field. Hence the field inside the box must be zero.
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