Download Lecture Notes: Y F Chapter 21

Electric Charge, Electric Forces, and Electric Fields (Y&F chapter 21)
1. “Ordinary” Matter is made up of elementary particles called:
electrons, protons, and neutrons.
2. These particles have mass and a property called “Electric Charge”
3. Electric Charge, or “charge” is a scalar quantity.
4. The electric charge of the electron is negative, the electric charge of the
proton is positive, and the neutron – zero.
5. The magnitude of the electric charge of electron is equal to that of the
proton.
Important Principle Number One:
Charge is Conserved - The algebraic sum of all the electric charges
in any closed system is constant
Important Principle Number Two:
The magnitude of the electron (and proton) charge is the natural unit”
of charge. Fractional charges are not observed in ordinary matter.
Important Observation About Charge
There are a huge number of of electrons, and protons in a
“human-sized” piece of matter (~1028 electrons in your body).
Your body is (approximately) electrically neutral because there are an
approximately equal number of protons and electrons in your body.
When we speak of a body as having a non-zero charge we usually
mean that it has an imbalance in the number of + and – charges.
Because the number of individual charges in a “macroscopic” object is
so large, we will often consider charge to be a continuous variable
(i.e. a real number) as opposed to being a discrete variable (i.e. integer)
This is similar to the situation where we identify the MASS of an object in terms of
a continuous variable (kilograms) when, in fact, it is actually given by the number of
atoms in in the object times the mass of an individual atom.
Coulomb’s Law
-There is a force between any two charges
-The force is directed along the line between
the charges
-The force is attractive if the charges are of
different sign, repulsive if they are the same.
-The magnitude of the force is proportional
to the product of the two charges (q1 x q2)
-The force is inversely proportional to the
square of the distance between the charges.
r
q1q2
F = k 2 r̂
r
Important - Force is a Vector
F (Force) is in
q (Charge) is in
r (Distance) is in
Newtons
Colombs
Meters
k must have units of Newtons(meters)2(coulomb)2
Coulomb’s Law
F (Force) is in
Newtons
q (Charge) is in
Colombs
r (Distance) is in
Meters
r̂ (Unit Vector- points AWAY from other charge)
r
q1q2
F = k 2 r̂
r
k must have units of
Newtons(meters)2/(coulomb)2
For reason that will become clearer soon, we choose to define the
constant “k” in terms of another number:
2
N
⋅
m
k=
= 8.988 x10 9
(4πε 0 )
C2
1
r
F=
1
q1q2
r̂
2
4πε 0 r
r̂
r
F=
1
q1q2
r̂
2
4πε 0 r
Principle of “Superposition”
tells us the force on a charge from a single other charge.
The principle of superposition states that the force on a charge from a
collection of other charges is just the vector sum of the individual forces from
each of the charges.
To Determine the force on charge q from a number of charges q1, q1, q1 …
r
F=
qq1
1 qq2
1 qq3
r̂ +
r̂ +
r̂ + ...
2 1
2 2
2 3
4πε 0 r1
4πε 0 r2
4πε 0 r3
1
r
1 qqi
F =∑
r̂
2
i 4πε 0 ri
The Electric Field
If I place a charge q in the vicinity of a collection of charges q1, q1, q1…then the
force on q will be given by :
r
1 qqi
F =∑
r̂
2 i
i 4πε 0 ri
Note that the Total Force is proportional to q:
r
1 qi
F = q∑
r̂
2 i
i 4πε 0 ri
or
r
r
F = qE
The Electric Field
Minor Problem…What if putting charge q in the vicinity of the other charges
causes them to move?
Definition of the Electric Field:
r
E = lim it q →0
r
F
q
We can calculate the Electric filed from a collection of charges from:
r
1 qi
r̂
F = q∑
2 i
i 4πε 0 ri
Which Gives
v
1 qi
r̂
E=∑
2 i
i 4πε 0 ri
Y&F Example 21.10
Example: Electric Field from a ring of charge
Step 1 – Consider a tiny portion of the ring (call it dQ) and treat
it as a point charge
r
Step 2 – Calculate the contribution to the total field (call it dE ) that results
from dQ.
Step 3 – “Add up” these tiny bits of electric by evaluating an integral:
r
r
E = ∫ dE
Note: Below is the sort of level of detail I would expect for a problem on an Exam
Question
What does the field from the ring of charge look
like if we are very far away?
1
xQ
Ex =
4πε 0 ( x 2 + a 2 ) 3 / 2
If x2>>a2 then x2 + a2 → x2
Ex =
Ex =
1
xQ
4πε 0 ( x 2 ) 3 / 2
1
xQ
4πε 0 x 3
1
Q
Ex =
4πε 0 x 2
This looks just like the field from a point charge
Question
What does the field from the ring of charge look
like if we are very far away?
1
xQ
Ex =
4πε 0 ( x 2 + a 2 ) 3 / 2
If x2>>a2 then x2 + a2 → x2
Ex =
Ex =
1
xQ
4πε 0 ( x 2 ) 3 / 2
1
xQ
4πε 0 x 3
Three Important Concepts:
1. Breaking a continuous distribution
into infinitesimal pieces and
integrating.
2. Using symmetry to simplify the
set-up of the problem.
3. Checking the result by looking at
the “limiting” behavior.
1
Q
Ex =
4πε 0 x 2
This looks just like the field from a point charge
v
1 qi
r̂
E=∑
2 i
i 4πε 0 ri
For a discrete set of point charges:
What happens if the charges are not points but are distributed continuously?
r
We can replace the charge qi by the “infinitesimal charge dq (r ) and write:
r
v
1 dq(r )
E=∫
r̂
2
4πε 0 r
all
r
If we say the charge is distributed by a charge density given by ρ (r ) we have:
r
r
dq (r ) = ρ (r )dv
This gives:
v
E=
dv is an “infinitesimal” volume element
r
ρ (r )
1
∫ 4πε
all
0
r
2
r̂dv
Don’t Panic – There won’t be any hard problems that use this!
The Electric Field
Minor Problem…What if putting charge q in the vicinity of the other charges
causes them to move?
Definition of the Electric Field:
r
E = lim it q →0
r
F
q
We can calculate the Electric filed from a collection of charges from:
r
1 qi
F = q∑
r̂
2 i
i 4πε 0 ri
Which Gives
or for continuous charge distribution
v
1 qi
E=∑
r̂
2 i
i 4πε 0 ri
v
E=
r
ρ (r )
1
∫ 4πε
all
0
r
2
r̂dv
ELECTRIC FIELD “LINES”
An Electric Field assigns VECTOR to each point in space:
r r
E (r3 )
r r
E (r1 )
r
r1
r
r2
r r
E (r2 )
r
r3
A convenient way to visualize the Electric Field is with “ELECTRIC FIELD LINES”
The local direction of the Field Lines is the direction of the
electric field at that point
The “density” of electric field lines is proportional to the
magnitude of the electric field at that point
The direction of the electric field line give the direction of the force
on a charge particle at that point. It does not necessarily represent
the direction of motion of a charged particle at that point!
Electric Field Lines
The electric field line is everywhere tangent to the local electric field
Electric Field Lines are CONTINUOUS Lines
That Begin and End on Charges
The Sense of the Field Lines is Away from the + charge
The Local Direction of the Electric Field is along the Field Line
The Local Strength of the Electric Field is Proportional to the Density of Lines
A Note About Conductors and Insulators
A CONDUCTOR is a material in which electric charges are free to “move around”
An INSULATOR is a material in which the electric charges are immobile
Conductors can be CHARGED or UNCHARGED
Insulators can be CHARGED or UNCHARGED
(refers to net charge)
Electric Field Inside a Conductor
1. Charges can move freely inside a conductor
(actually it is only the electrons that move)
2. Assume that there is an Electric Field Inside Conductor:
3. Since the Charges are free to move, they we be pushed towards the
surface of the Conductor:
4. The field of these charges will tend to cancel the original Field.
5. They will continue to move until the field is zero
The Electric Field inside a Conductor is EXACTLY ZERO
End of Chapter 21
You are responsible for the material covered in T&F Sections 21.1-21.6
You are expected to:
•
Know what the following are:
•
Understand Coulomb’s Law including the notion of a unit vector and to be able to
express the direction of the electric field.
•
Calculate the force on a point charge due to another point charge.
•
Calculate the electric field from a point charge and, using superposition, calculate the
electric field from a simple collection of point charges.
•
Understand the notion of Electric Field Lines and be able to roughly sketch the field
lines for simple collections of charges.
charge, electrons, protons, atomic number, conductors, insulators, electrostatic force,
superposition, and electric field.
Recommended F&Y Exercises chapter 21:
3,13,14,15,16,24,44,49