Download Physics 272: Electricity and Magnetism

Physics 272: Electricity and
Magnetism
Mark Palenik
Thursday June 21st
Topics
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Review of potential energy
Electric potential
Potential due to charges
Over the next few days, develop more details
about how potential/potential energy works
Energy
• Full relativistic energy
𝐸=
𝑚𝑐 2
+𝑈
Potential
1 − 𝑣 2 /𝑐 2
= 𝑚𝑐 2 + 𝐾 + 𝑈
• mc2 doesn’t change (usually), and K ~ ½mv2, so for our
purposes:
• Two kinds of energy: kinetic and potential
1
𝐸 = 𝐾 + 𝑈 ≈ mv 2 + U
2
• Energy is conserved if no external work is done
Energy Continued
• All of physics can be done with energy instead of force
– Quantum mechanics can only be done this way (it seems)
• You’ve learned a basic version of this: conservation of
energy
 ∆𝐸 = 𝑊 the change in energy of the system = external work
done on the system
 If there is no external work, kinetic+potential = constant
 To use conservation of energy, we must consider kinetic
energy of charges and potential energy
Potential/potential E, work
• Electric potential, potential energy, and work
are all related
• So, let’s review work
– This leads to idea of potential energy
– This, then leads to electric potential
Work
• Remember, work is done by external forces
• Work changes the energy of a system
• 𝑊=
𝐹 ∙ 𝑑𝑥
dx
F
dx
A particle moves along a curvy wire, but the external force always pushes to
the right
iClicker –What does this mean?
• What does it mean when we say W =
𝐹 ∙ 𝑑𝑥
a) Only the motion of the particle PERPENDICULAR to
the force contributes to the work done on it
b) Only the motion of the particle PARALLEL to the force
contributes to the work done on it
c) Both PARALLEL and PERPENDICULAR components
contribute to the work
Work review (clicker)
• A horizontal force of 10 N pushes a bead along a wire. The wire
has a length of 25 m. The horizontal displacement of the bead
when it reaches the end of the wire is 10m. The vertical
displacement is 1m. How much work was done moving the
bead?
L=25m
dx
F=10 N
a)
b)
c)
d)
10 J
250 J
100 J
100 N
Dy = 1
Dx=10 m
Potential energy of charges
• Remember: potential energy comes from interaction of
TWO objects
• We can find potential energy by checking the
interaction of 2 particles
q1
q2
Hold q1 fixed, move q2. How much work
do we have to do?
iClicker
• We wish to find the work that it requires to move q2
along the horizontal path away from q1. Which F
appears in the expression 𝑊 = 𝐹 ∙ 𝑑𝑥 ?
q1
q2
a) The force exerted on q2 by q1, Fq2q1
b) The force need to counteract the force
exerted on q2 by q1, - Fq2q1
c) Any constant, horizontal force
Work to move a particle
𝑏
𝐹
𝑎
• 𝑊=
∙ 𝑑𝑥 in this case represents the energy
required to move q2 from a to b at constant speed
a
q2
r
q1
b
If the force between q1 and q2 is attractive, we must pull
the particles apart.
If it is attractive, we must push in to keep it from
speeding up
• 𝐹=
𝑞1𝑞2
−
𝑟
4𝜋∈0 𝑟 2
and w =
𝑏
𝐹
𝑎
∙ 𝑑𝑥 = −
𝑞1𝑞2
𝑞1𝑞2
𝑊=
−
4𝜋 ∈0 𝑟𝑏 4𝜋 ∈0 𝑟𝑎
𝑏 𝑞1𝑞2
𝑟
𝑎 4𝜋∈0 𝑟 2
∙ 𝑑𝑟
Angular motion
• We’ve shown what the work is required to
move a charge along the radial direction.
• The work required to move a charge in the
angular direction (to circle q2 around q1) is
zero.
• This is because we are moving perpendicular
to field lines.
• We’ll discuss this in greater detail in next
lecture.
What happened to the energy?
• We did work on the system, but kinetic energy didn’t
change (moved particle at constant speed)
• Work ALWAYS implies a change in energy
– Where did it go?
– Potential energy!
• The change in potential energy is:
Same as
work
∆U =
𝑞1𝑞2
4𝜋∈0 𝑟𝑏
−
𝑞1𝑞2
4𝜋∈0 𝑟𝑎
• This is Potential energy at b – potential energy at a
• Therefore potential energy is:
𝑞1𝑞2
U=
4𝜋 ∈0 𝑟
iClicker quiz
• Two particles with charge q sit a distance d
apart. What is the potential energy of the
system, including both particles?
q1
a)
b)
c)
d)
2q1q2/4pe0d
q1q2/4pe0d
2q1q2/4pe0d2
q1q2/4pe0d2
d
q2
Energy/charge
• We have the potential energy of two charges
interacting, we can define the electric potential
• Electric potential doesn’t belong to a single charge. No
potential energy until a second charge is added
• Potential is energy per charge
q
• Potential is: 𝑉 =
𝑞
4𝜋∈0 𝑟
r
= − 𝐸 ∙ 𝑑𝑥
• This means that the potential, V uniquely defines the
electric field of an object (next lecture)
Change in KE
• The electric field from two plates with charges +Q and
Q
–Q is
𝑥 What is the change in a proton’s kinetic
𝐴∈0
energy as it moves from A to B?
• What are two ways we can look at the
problem/system?
Rules for potential
• Remember: Field lines point away from positive
charges
• Field lines point in direction of DECREASING potential
• Therefore: Potential increases as you move toward
positive charges or away from negative charges
• Electric potential of a charge is
𝑞
𝑉=
= − 𝐸 ∙ 𝑑𝑥
4𝜋 ∈0 𝑟
• Potential energy of two charges interacting is
𝑞𝑞2
• U=
= −𝑞2 𝐸1 ∙ 𝑑𝑥 = −𝑞1 𝐸2 ∙ 𝑑𝑥
4𝜋∈0 𝑟